Zoomorphologie 91, 49-61 (1978)

Zoomorphologie O by Springer-Verlag 1978

Mechanical Analysis of the External Forces on Climbing Mammals Walter J. Bock 1, and Hans Winkler 2 Department of Biological Sciences, Columbia University, New York, NY 10027, USA 2 Institut ffir Limnologieder OsterreichischenAkademie der Wissenschaften, Berggasse 18/19, A-1090 Wien, Austria

Summary. Analysis of external forces acting on a climbing mammal under static conditions can be made with the method of free-body diagrams which requires the minimum number of measurements. The analysis must satisfy simultaneously equilibria for both linear translational and rotational effects of the external forces. Solution of the equations generated from the free-body diagram will give the vector direction and magnitude of the contact forces acting on the animal at the supports. Determination of these external forces cannot solve all aspects of climbing adaptations; it provides only the first essential step.

A. Introduction Analysis of the external forces acting on a climbing animal, be it a primate, other arboreal mammal, bird or any other animal including invertebrates, can be made using the same physical method because the consequences of these forces on the animal are basically similar. Once a proper physical treatment of the external forces is available, it would then be possible to inquire into the nature of the forces acting within the animal's body, e.g., stresses on bones and the muscular forces required to hold the bones in position. Further, it would be possible to ascertain which morphological features of primates are adaptations for climbing (e.g., Cartmill, 1974) and to determine which measurements must be made to judge the relative merits of these adaptations. Finally, it would be possible to compare in a meaningful way the diverse adaptations (i.e., the different paradaptations) for climbing in primates and other groups of arboreal mammals (see Bock, 1977, for a discussion of comparison and adaptation). In a recent symposium volume on "Primate L o c o m o t i o n , " Badoux (1974) and Cartmill (1974) presented two different mechanical analyses of the external forces acting on mammals clinging or climbing on upright surfaces. *

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w.J. Bock and H. Winkler

They discussed various morphological features serving as adaptations for climbing in primates and other arboreal mammals, and compared some of the consequences of evolutionary modifications of these features. No discussion was presented by either author on the compatibility of their methods, i.e., whether either reduces to the other, or whether both are valid. Nor did the authors present clearly the complete set of assumptions on which their analysis was based or discuss the minimum number of measurements required for a complete solution of the external forces acting on the animal. Our interest in these physical models stems from our earlier treatment of the external forces acting on climbing woodpeckers and other birds (Winkler and Bock, 1976); our model differs from both those advocated by Badoux and by Cartmill. A comparison of the three treatments indicated that neither the Badoux model nor the Cartmill model could be reduced to the other nor in any simple way to the one we advocated which suggests that one or more may be based on incomplete or erroneous assumptions, or that the minimum number of required measurements needed to complete the analysis differs. We would like to present a general model of the external forces acting on a climbing mammal based upon our analysis of climbing birds and to use this model to analyze the validity of the earlier ones advocated by Badoux and by Cartmill. The model is a general one that can be used for any climbing animal with the only modification being the number of terms in the equations and the minimum number of required measurements to be made.

B. The General Model

In this analysis we will consider only static situations in which the climbing animal is suspended motionless from the support. For simplicity, the animal will be treated as a perfectly symmetrical " t w o dimensional" organism seen in lateral view. This assumes that the forelimbs and the hindlimbs are placed symmetrically on the supports and that the forces acting on each forelimb are equal as are those acting on each hindlimb. This assumption need not be made, but the physical model and analysis of unsymmetrically placed limbs becomes unnecessarily complex and cumbersome for the purposes of a general presentation. In all diagrams, the forces shown on the forelimb and on the hindlimb are the total force acting on both members of the paired limb. The indicated value must be divided in half to obtain the actual force acting on each forelimb or on each hindlimb. The model is valid for mammals climbing on supports at all positions relative to the horizontal, be they sloping (~1 greater than 90~ vertical (~1 =90~ or overhanging (~1 less than 90~ We chose the last for our general analysis because it emphasizes the rotational effect of the external forces without being a special case. Our model will be for mammals supporting themselves with symmetrically placed forelimbs and hindlimbs. It could be extended to fewer supports with reduction in the number of terms and required measurements or to more supports with increase in the number of terms and required measurements. Thus, mammals supporting themselves by two pairs of limbs and a

Mechanical Analysis of the External Forces on Climbing Mammals

51

prehensile tail, or by any other combination of supports, could be analyzed with the proposed model provided that the proper minimum number of measurements are made. In almost all cases, a climbing mammal regardless of the position of the support, will be subjected to the rotational consequence as well as the linear translation of the external forces acting on it. One exception is when the animal is hanging by one limb or a prehensile tail from a horizontal support. Thus the external forces acting on the animal place torques or moments on it and tend to rotate it. If the mammal is in a static condition then two conditions of equilibrium, that of rotational equilibrium and that of linear transitional equilibrium, must be satisfied. An analysis of the external forces acting on the animal, to be valid and complete, must contain equations showing that the sum of the torques acting on the animal is equal to zero and showing that the vector sum of the linear forces acting on the animal is equal to zero. Generally three simultaneous equations are needed, one for the sum of the moments (torques) about some arbitrarily chosen center of rotation and two for the sum of linear components of the forces acting on the body. The vector summation of the linear forces could also be done graphically if sufficient information was known about these forces; usually the information to permit graphic methods is lacking. The linear forces will behave as if the entire mass of the animal is concentrated at its center of gravity and the forces acted at this point. A complete analysis of the external forces is provided by the method of free-body diagrams (see Dempster, 1961; Bock, 1974). A great advantage of this method is that it necessitates identification of all external forces acting on each body (i.e., the climbing mammal) and that it permits a complete analysis with the minimum number of measurements. The last is especially valuable in biomechanics because of the difficulties in obtaining many of the necessary measurements in actual studies. A major problem for the analysis of the external forces acting on a climbing animal is that the nature of the contact forces between the mammal and its external supports is not known at the o n s e t - t h a t is, neither the direction nor the magnitude of these forces are known, nor can they be guessed prior to the completion of the analysis. Thus the vector direction of the contact forces cannot be obtained from a knowledge of only the structural relationships of the animal and its supports. These contact forces are not necessarily perpendicular to the curvature of the contact surface (the assumption of Badoux, 1974) nor do they necessarily lie parallel to the longitudinal axis of the limbs or tail (the assumption of Stolpe, 1932). The difficulty stems from the existence of frictional forces acting between the animal and the substrate which depend upon the coefficient of friction (usually unknown) and the magnitude of the perpendicularly directed forces which are generally not known until completion of the analysis. A general model for the external forces acting on a climbing mammal can be constructed as shown in Figure 1. The entire animal is considered as a single free-body; we are not concerned with any internal forces, muscular or otherwise, acting within the animal's body. Because the vector direction of

52

W.J. Bock and H. Winkler

Fig. 1. Free-body analyze of a mammal at rest clinging to an oblique (overhanging) surface-the general model. The supports of the mammal on the tree are at its forelimbs (c) and its hindlimbs (o). The force of gravity is acting at the animal's center of gravity. The analysis will solve for the unknown contact forces, FI and Fh, with the minimum number of measurements; see the text for detail. The three forces acting on the animal form a closed polygon of forces

s o m e o f the e x t e r n a l forces on the a n i m a l d o n o t pass t h r o u g h its center o f gravity, the a n i m a l will t e n d to be r o t a t e d as well as m o v e d in a linear direction. Hence, with the basic a s s u m p t i o n t h a t the a n i m a l is u n d e r static c o n d i t i o n s , the sums o f the m o m e n t s a n d o f the linear t r a n s i o n a l effects o f the forces m u s t b o t h be e q u a l to zero. T h e p h y s i c a l factors which are k n o w n , by p r i o r m e a s u r e m e n t , at the o n s e t are: (a) P o i n t s o f c o n t a c t o f the h i n d f e e t a n d o f the forefeet on the tree, (b) P o s i t i o n o f the center o f gravity o f the a n i m a l ' s b o d y , (c) Size (weight) a n d d i r e c t i o n (vertical a n d d o w n w a r d ) o f the force o f gravity, a n d (d) Slope o f the s u p p o r t i n g surface with respect to the h o r i z o n t a l (c~1 in Fig. 1). S a t i s f a c t i o n o f the c o n d i t i o n s o f linear e q u i l i b r i u m is given by the f o l l o w i n g equations: F1 -+-F2 -- F6 = 0

(1)

F 3 + F 4 = 0,

(2)

Mechanical Analysis of the External Forces on Climbing Mammals

53

and satisfaction of the condition of rotational equilibrium is given by the following equation:

--FG (oa) +F1 (ob) +F3 (oc) =0.

(3)

Equation (1) indicates that the vertically directed force of gravity acting at the center of gravity is balanced by two vertically directed forces acting at the contacts of the forelimb and of the hindlimb on the tree. The other component may be represented by a pair of parallel, equal and opposite forces, F3 and F4, which are placed, for convenience, perpendicular to the surface of the tree; these do not change the translational effect of the external forces on the animal. The contact between the hindfoot and the tree has been chosen as the center of rotation for the analysis. Because the center of rotation can be arbitrarily placed at any point, this one was chosen for convenience to the analysis. Moment arms from this center of rotation, oa, ob, and oc, are drawn to each torque producing force (i.e., those not passing through the chosen center of rotation), FG, Fa, and F3, acting on the animal. The forces represented in equation 1 as well as those represented in Equation (2) are parallel to each other and hence can be added in a simple scalar fashion. Equations (1) and (2) represent the equilibrium for the linear translation effect of the forces and Equation (3) represents the equilibrium for the rotational effect (the torque or moment) of the forces. All of these equations must be equal to zero to satisfy the condition of static equilibrium. The following equations should be given for completeness of the model, namely: cz1 ~- ~2 ~-- 9 0~

(4)

F 1 =Fy/cos

~2,

(5)

F2 = F J c o s c~2.

(6)

and

The forces, Fy and Fz, are ones acting on the forefoot and the hindfoot respectively in a direction parallel to the surface of the tree. Hence these forces are perpendicular to forces F3 and F4. It is necessary to introduce forces, Fy and Fz, and to determine their relationships via Equations (5) and (6) to the vertical forces, F1 and F2, because the latter forces cannot be measured directly. Because of the way the forces were chosen in the free-body diagram, the vertical forces include a contribution from the forces acting perpendicular to the tree which cannot be eliminated in any simple way. Hence the vertical forces cannot be measured directly. Rather the force parallel to the tree surface must be measured and converted to the vertical force via Equation (5) or (6); forces Fy and F z do not contain any contribution from those acting perpendicular to the tree surface.

54

w.J. Bock and H. Winkler

Solution of the three simultaneous Equations, (1), (2), and (3), generated by the free-body diagram requires a minimum of the following measurements or of a similar combination: (a) Moment arms oa, ob, and oc (or measures that can be converted to these moment arms), (b) Angle el or ~2 [see Eq. (4)], (c) Fy or F~ [gives F1 or F2, see Eqs. (5) and (6)], and (d) F6. This list represents the easiest set of physical values to be measured in any actual study. The most difficult measurement to obtain would be F y or F z for which a force plate with an appropriately positioned strain gage must be placed at the contact of one of the feet. Given values for these measurements, Equations (1), (2), and (3) can be solved. With the solutions for forces FI and /73, and for forces F2 and F4, the actual forces Fy on the forefoot and F h on the hindfoot can be obtained by a simple vector summation (parallelogram of forces) as shown in Figure 1. Thus the forces, F-C and Fh, are the actual forces exerted by the tree on the animal at the two points of contact. The force at the forelimb, F-C, is a tensile force of the tree on the animal and in this case is resisted by sinking the claws of the forefoot into the tree. The force of the hindlimb, F h, is a compressive force of the tree on the animal and in this case is resisted by friction between the pad of the foot and the surface of the tree. As the animal is in a condition of static equilibrium, the three external forces acting on it F6, F c, and F h, will form a closed polygon of forces as shown in Figure 1. In the example shown in Figure 1, a value of 100 units, has been assigned to FG and one of 60 units to F 2. The lengths of the moment arms have been measured directly from the diagram. Solution of the equations generated from the free-body diagrams gives forces F h = 6 0 units and F-C=63 units at the two contact points with vector directions as shown in Figure 1. The nature of the support, whether sloping or vertical, and the number of contact points and hence forces (e.g., hands only or the use of a prehensile tail) will alter the free-body diagram and the equations (number of terms and perhaps the signs of the terms), but the physical analysis is basically the same. Most essential is the requirement for equilibrium of both the linear translational and the rotational effects of the external forces. The type of contact forces between the mammal and its supports, whether these are tensile or compressive forces of the support on the animal, and the method by which the mammal resists these forces, be it by sinking its claws into the substrate, by friction of the foot pad or whatever, depends upon the animal, the nature of the substrate and how the animal climbs. Most important is that the analysis should show the force, its vector direction and magnitude, of the substrate on the animal at each contact point. The advantage of a free-body diagram is that solution of the associated equations will give the external forces that act on the mammal be these contact or at a distance ( = gravity) forces, with a minimum number of required measurements.

Mechanical Analysis of the External Forces on Climbing Mammals

55

IlO

Fig. 2. The Badoux model taken from Badoux (1974, p. 26, Fig.23). The contact forces are those exerted by the animal on the supports

C. The Badoux Model

Badoux (1974, Fig. 23, p. 24) presented a model for the distribution of the forces generated by the animal's weight for a vertically climbing primate. The contact forces shown are those exerted by the animal on the support and hence are equal and opposite to those indicated in a free-body diagram. His analysis (Fig. 2 copied directly f r o m Fig. 23 of Badoux) is based on the tacit assumptions that the contact forces between the animal and its supports are normal (perpendicular) to the curvature of the support surface, that the projection of the force vectors can be continued to their point of intersection and that a parallelogram of forces with the vertically directed force of gravity (the animal's weight) as the diagonal can be constructed at this point. The last two assumptions are valid and will provide a correct solution given the proper placement of the vectors of the contact forces. The central assumption underlying the Badoux model is that the contact forces are perpendicular to the curvature of the support surface at the point of contact with the animal. This assumption ignores the rotational effect of the forces acting on the animal and hence the resulting frictional forces at the contacts, and hence is not valid. The vector directions of the contact forces in the Badoux model, as shown in Figure 2, are incorrect because they satisfy only the condition of translational equilibrium, not both conditions of translational and rotational equilibria. The parallelogram method used by Badoux

56

W.J. Bock and H. Winkler

to obtain the magnitude of the contact forces is valid, but the solutions for these forces obtained in his analysis are incorrect because of the erroneous assumption for the vector directions of these forces. Badoux's model is similar, but not identical, to that advocated many years ago by Stolpe (1932) for woodpeckers. The major difference is that Stolpe places the vector direction of one contact force along the longitudinal axis of the tail and constructs the force parallelogram at the center of gravity with rectilinear force components. However, they depend on the same basic assumptions, and the objections raised against Stolpe's model (Winkler and Bock, 1976, pp. 398-399) apply equally well to Badoux's model. The major objections are (a) that the rotational consequences of the external forces on the animal's body must be included in the analysis, and (b) that the vector direction of the contact forces cannot be ascertained from the morphology of the model, e.g. placed perpendicular to the curvature of the supporting surface as in this model. A reanalysis of the vertically climbing macaque drawn exactly from Badoux (1974, Fig. 23) is presented in Figure 3 following the general analysis given above. The weight of the animal was assumed to be 100 units and the vertical force on the hindlimb, F2, to be 60 units. The lengths of the moment arms were measured directly from the diagram. Solution of the free body equations gives values for the contact forces of 79 units for Fj. and of 90 units for F h. These differ both in magnitude and vector direction from those shown in Badoux's analysis. The two solutions are not comparable because we assigned the value of 60 units to force F2 arbitrarily and do not know the correct value of this force which must be measured. Note that in our analysis the contact forces are not normal to the curvature of the contact surface.

l ~=~

N

Fig. 3. The Badoux example reanalyzed by the method of freebody diagrams. The contact forces are those exerted by the supports on the animal

Mechanical Analysis of the External Forces on Climbing Mammals

57

v 9 ,Ff =

42

/'F:3 = 16.5

I00

I//UI

,'

i'

t I

// Ill"

d .p

) .l[

4

Fig. 4. The Cartmill model taken from Cartmill (1974, p. 69, Fig. 6). The forces shown at A and B are rectilinear components of the contact forces of the support on the animal Fig. 5. The Cartmill example reanalyzed by the method of freebody diagrams. The contact forces are those exerted on the animal by the support

The model advocated by B a d o u x (1974) is not valid because it does not satisfy simultaneously b o t h conditions of transactional and rotation equilibrium.

D. The Cartmill Model

Cartmill (1974, Fig. 6, pp. 59-61) also presented a model for the forces acting on a vertically climbing m a m m a l (see Fig. 4 copied f r o m Cartmill's Fig. 6). He clearly includes the rotational consequence of the animal's weight in his analysis but uses a set of equations that is quite different in appearance f r o m that in the general model advocated above. Cartmill's two equations are (see Fig. 4):

Wd + Avt= Ahk,

(7)

W(d + t) = B J + Bhk.

(8)

and

Each equation is a sum o f moments, but each is a b o u t a different center of rotation. Cartmill concludes that these equations hold when the animal is in

58

w.J. Bock and H. Winkler

a state of static equilibrium. Note that these equations are not a pair of simultaneous equations; insufficient connections are provided between them as the Av and A h terms differ from the By and B h terms. They are alternate and redundant ways of describing rotational equilibrium of the external forces on the animal. Cartmill is correct in his conclusion that when the climbing primate is in static equilibrium, these two equations (//7 a n d / / 8 ) will hold. But he presents his model in the form of an analysis by which the direction and magnitude of the contact forces can be ascertained. As such, no instructions are provided on how the analysis should be carried out and in particular on the minimum number of measurements essential for the analysis. These details are needed because Cartmill (1974, pp. 59~61) uses the model to discuss the consequences of varying several factors, such as the position of the center of gravity relative to the tree or the closeness of the contact points, as the basis for analysis of the adaptiveness of various morphological features (e.g., pads and claws) for vertical locomotion. The reanalysis of Cartmill's example using the free-body diagram approach advocated above is given in Figure 5. The weight of the animal was assumed to be 100 units and the vertical force on the hindlimb, F2, to be 60 units. The lengths of the moment arms were measured directly from the diagram. Solution of these equations gives values for the contact forces of 42 units for FI and of 64 units for F h. Cartmill did not provide figures for an actual example in his paper, but we substituted the same values for the force of gravity, the vertically directed forces and the moment arms, and solved for the equations given by Cartmill. The same values are obtained for the contact forces at the forelimb and at the hindlimb with both analyses. Cartmill's method can be reduced to that of the free-body diagram and is hence a valid one. Only one of his two equations for rotational equilibrium is needed (the other is redundant), and to that equation must be added the two equations (using his terms) needed to describe translational equilibrium, namely: A h - Bh = 0,

(9)

and A~+B~-W=O.

(10)

Use of these equations permits elimination of either one of the two rotational equations in Cartmill's model with the remaining three equations forming the set of simultaneous equations requiring the fewest measurements for a complete solution of the external forces acting on the animal. Although Cartmill's method is valid and can be reducted to the method of free-body diagrams, it is not strictly equivalent to the latter. The major difference between Cartmill's approach and that of free-body diagrams is that m o r e p h y s i c a l v a r i a b l e s m u s t be m e a s u r e d in the Cartmill method because his equations are nonsimultaneous. It is necessary to measure simultaneously one of the A forces (either A~ o r Ah) and one of the B forces (either

Mechanical Analysis of the External Forces on Climbing MammaIs

59

By or Bh) in the Cartmill method, but only F 1 or F 2 ( = A v or By) using the method of free-body diagrams. This is a very important difference because of the experimental difficulties of measuring an A force and a B force simultaneously. Construction and use of an experimental device with two force plates positioned properly so that these two forces can be measured simultaneously for a climbing m a m m a l i s far more difficult than the construction and use of a device with a single force plate. Hence the method advocated by Cartmill is cumbersome because it requires the measurement of more variables than does the free-body diagram. The discussion by Cartmill (1974, p. 59) that the weight of the animal produces rotatory moments about two pairs of axes because the center of gravity is not coplanar with the points of support is confused. Several moments of force may act on the animal; this is simply a consequence of the pattern of forces acting on it. But to find the sum of these several moments, one point of reference must be designated, not several as used by Cartmill. Vector summation of moments is made by arranging these moments perpendicularly (plus and minus) to the plane of rotation at the chosen center of rotation as done in equation 3. The solution to the analysis will be exactly the same regardless of which point is taken as the center of rotation. We wish to emphasize that our disagreement with Cartmill is with the usefulness and ease that his method of analyzing external forces can be applied to an actual case as compared with the method of free-body diagrams. We are in agreement with most of the conclusions presented by Cartmill (e.g., p. 61) and with the reasons on which these conclusions are based.

E. Discussion

It is not possible to formulate a general conclusion on the best position for a climbing m a m m a l or other arboreal animal to hold its limbs (limbs symmetrical) to minimize the forces because of the different ways these forces are resisted by the feet, e.g., claws or friction soles. As pointed out by Cartmill (1974, p. 61) and ourselves (Winkler and Bock, 1976) it is not valid to conclude that it is always advantageous for a climbing primate to press the center of gravity as close as possible to the supporting surface. N o r is it always advantageous to minimize or to maximize the distance between the two supports. Perhaps the only general conclusion is that when an animal is subjected to rotational effects of the external forces (i.e., some forces do not pass through the center of gravity), then the sum of the contact forces is always greater and sometimes much greater than the weight of the animal (the force of gravity). However, this conclusion is not very interesting or useful for further investigation. The analysis of external forces acting on a climbing m a m m a l by the method of free body diagrams can solve only the question of the nature of these forces, i.e., their magnitude and vector direction. It cannot provide information on the nature of forces within the body of the animal, on how the contact forces are resisted by the animal, on what features may be adaptations for climbing, and on the relative merits of various adaptations for climbing. Solution of

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W.J. Bock and H. Winkler

these problems requires methods other than that of free-body diagrams, but these solutions depend upon information provided by the application of freebody diagrams. It is beyond the intent of our analysis to delve into the question of adaptations for climbing in various groups of mammals. It was our hope to provide the means by which one of the initial steps may be taken correctly.

F. Conclusions

A) The nature of the contact forces, their vector direction and magnitude, acting on a climbing mammal or other arboreal animal may be ascertained with the use of free-body diagrams; this method requires the minimum number of measurements. If any of the external forces acting on the animal do not pass through its center of gravity, then the simultaneous equations derived from the diagram must satisfy equilibria for linear translational and for rotational effects of the external forces. Generally one equation for the sum of the moments and two equations for the vector summation of the linear forces are needed. B) The vector direction of the contact forces from the supports on a climbing mammal cannot be ascertained from the morphological relationships between the animal and its supports because of the possible existence of frictional forces at the contacts. C) If one or more external forces on a climbing mammal does not pass through its center of gravity, then the sum of the contact forces will be greater than the force of gravity ( = the animal's weight). If the animal is in static equilibrium, the external forces will form a closed polygon of forces. D) The analysis presented by Badoux is not valid because it does not satisfy rotational equilibrium. The analysis by Cartmill uses two non-simultaneous equations; it is valid but cumbersome because it requires an additional difficult measurement. Its use is not recommended. E) Free-body diagrams are suitable only for ascertaining the external forces acting on a climbing mammal. They are not suitable for solving for the internal forces or for determining which features are adaptations for climbing. Acknowledgements. We

wish to thank Taru Suzuki for drawing the illustrations. This study was done with support of a research grant, N S F - B N S 7306818 from the National Science Foundation to Walter Bock and a Frank C h a p m a n grant from the American M u s e u m of Natural History to Hans Winkler; both supports are gratefully acknowledged.

References Badoux, D . M . : An introduction to biomechanical principles in primate locomotion and structure. In: Primate locomotion (F.A. Jenkins, jr., ed.), pp. 1 4 3 . New York: Academic Press 1974

Mechanical Analysis of the External Forces on Climbing Mammals

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Bock, W.J.: The avian skeletomuscular system. In: Avian biology (D.S. Farner, J.R. King, eds.), Vol. IV, pp. 119-257. New York: Academic Press 1974 Bock, W.J.: Adaptation and the comparative method. In: Major patterns in vertebrate evolution (M.K. Hecht, P.C. Goody, B.M. Hecht, eds.), pp. 57-82. New York: Plenum 1977 Cartmill, M. : Rethinking primate origins. Science 184, 436443 (1974) Cartmili, M.: Pads and claws in arboreal locomotion. In: Primate locomotion (F.A. Jenkins, jr., ed.), pp. 45-83. New York: Academic Press 1977 Dempster, W.J.: Free-body diagrams as an approach to the mechanics of human posture and motion. In: Biochemical studies of the skeletomuscular system (F.G. Evans, ed.), pp. 81-135. Springfield, Illinois: Thomas 1961 Stolpe, M.: Physiologisch-anatomische Untersuchungen fiber die hintere Extremit/it der V6gel, J. Ornith. 80, 161547 (1932) Winkler, H., Bock, W.J. : Analyse der Kr/ifteverhfiltnisse bei Kletterv6geln. J. Ornith. 117, 397-418 (1976)

Received April 13, 1978

Note Added in Proof After our paper was accepted we saw the recent study by C. Niemitz [" Zur funktionellen Anatomie der Papillarleisten und ihrer Muster bei Tarsius bancanus borneanus Horsfield, 1821 ". Z. S/iugetierk. 42, 321-346 (1977)] in which he presents a mechanical analysis of the forces acting on Tarsius while the animal is clinging on vertical tree stems (see his Fig. 2, p. 323). Niemitz's analysis is incomplete because it does not satisfy the equilibria of either the linear translational or the rotational effects of the external forces acting on the animal. His analysis as shown in his Figure 2 is only for the hind foot and ignores the contact forces acting on the front foot and on the tail. It is not possible, on the basis of Niemitz's analysis, to ascertain the position and magnitude of compressive forces and of shearing forces acting at the several contact points between the animal and the tree. Therefore, support is lacking for his conclusion that different patterns of epidermal ridges develop on contact pads subjected to compressive forces in contrast to those subjected to shearing forces.