324 



ANIMAL MECHANICS. 



Since the planes are at right angles, we have 



P 2 = 7T + 7T 2 + ll 2 , 



and, by equation (78), 



Tnr' = maximum, 



Differentiating these two equations, we find 



irdir + it'dir' + udil = o, 

 irdw + Tr'dir = o. 



Eliminating c?7r', we obtain 



(it 2 - 7T /2 ) drr + 7rw<ii« = o. 



This equation of condition is satisfied by making 



7T = 71-', 



11 = o; 



or, the points a and «' must coincide, and the perpendiculars 

 7T, 7r', must be equal to each other. In other words — 



The fibre PP' must lie in a plane perpeiidicidar to both the 

 intersecting planes, and its extremities must be equidistant from 

 their intersection. 



By means of this proposition, we are enabled to solve a 

 number of problems, suggested by the adductor muscles of 

 many animals. 



If the muscle be triangular, then the resultant of all its 

 fibres lies in the bisector of the vertical angle, which bisector 

 is of a given length, and may be regarded as a single fibre, 

 which is to be placed between the intersecting planes in con- 

 formity with the foregoing proposition. 



If the muscle be a plane quadrilateral, we are then given 

 two bones, ^i?and A'B\ which lie in two planes intersecting 



