ANIMAL MECHANICS. 



247 



Hence, we have 



F 2ak sin Odd 



•> 2 = 



2F_ 

 1 



I V a? + & - 2ctk cos 0 

 s/a?~T~k z - 2ak cos 0 + const. 



to = 



(Xi -x 2 ). 



= &. (45) 



where &c denotes the total change of length of OX in 

 passing from the angle Qi to the angle 0 2 . Hence the 

 angular velocity of the arm produced by the contraction of 

 the latissimus dorsi varies as the square root of that contrac- 

 tion ; and it is important, being supposed given the amount 

 of the contraction, to find the conditions necessary to produce 

 the resulting angular velocity with a minimum angle of 

 rotation. This may be thus determined, 



x % = a 2 + k 2 - 2ctk cos 0. 



Hence, 



xSx = ak sin 0<$0, 



x%x Sx 



CU = — 7~ 



ak sin 0 k sin <p 



Hence, W will be a minimum when sin 0 is a maximum, or 

 when <f> = 90 0 , which coincides with our previous result. 



When the arm is depressed 48 0 23' below the horizontal 

 line, so as to make the angle XOS a right angle, the length 

 of OX is 12.62 inches; and if the total contraction of the 

 muscle be one-eighth part of its length, we have for the 

 minimum angle of rotation, before and after passing through 

 the position of the arm, where XOS is a right angle, 



