2 3 



ANIMAL MECHANICS. 



In the method of polar co-ordinates we require to find the angle bop, 

 which is usually denoted by 6. This is regarded as plus, if it be measured 

 round the point o from a reference line in a direction the reverse of the 

 movement of the hands of a watch, and minus if in the same direction. The 

 radius vector op, denoted usually by r, must also be known ; it is always 

 taken as positive. 1 



If we simply require to determine an angular displacement of a limb, we 

 proceed as follows. A certain position of the limb, say the vertical, as in 



Fig. 120, is assumed 

 as the primary posi- 

 tion. (The move- 

 ments are assumed 

 to be confined to 

 the plane of the 

 paper.) The forearm 

 is then rotated for- 

 wards to V, and the 

 angle 



<f> gives its 

 new position in re- 

 lation to the primary 

 position from which 

 it started. In the 

 same manner the 

 angle x gives the 

 new position of the 

 forearm in relation 

 to the same vertical 

 line (ab). The same 

 may more usefully 

 be expressed by 

 taking </> as the angle 

 at which the shoul- 

 der is bent, and \p 

 the bending of the 

 forearm upon the arm. One may readily, however, pass from one method of 

 expressing the position to the other, for the angle \j/ = if/ and the angles \p, <f>, 

 and x are simply related to one another ; for 



x = l }, + <j> 



If the angle <£ be a rotation forwards, say of 30°, we may call this a negative 

 rotation of 30°, and we must in that case term an extension at the shoulder 

 backwards to ab" as a rotation of + 30°. 



So far we have limited our attention to the description of the posi- 

 tions or changes of position of points in space for two dimensions, but 

 the shoulder-joint can move on more than one axis, and in more than 

 one plane. In fact, to denote most of the changes of position of the 

 limbs which are brought about by muscular contraction, and to denote 

 completely the position in space occupied by various parts of the body, 

 we have to deal with a problem in tridimensional space, and must refer 

 these positions to three co-ordinate planes at right angles to each other. 

 In Fig. 121 these three planes are represented. The horizontal plane or 

 floor is that through which the lines yy and xx pass. One vertical plane 

 is that through which the lines yy and zz pass ; the other vertical plane 



1 It is quite immaterial which direction is considered as plus and which as minus, so long 

 as this is once settled and adhered to. 



Fig. 120. — This figure indicates the method of measuring the 

 angular displacement of the arm from a primary vertical 

 position (ab). 



