ANIMAL MECHANICS. 



241 



it is required to find the conditions necessary to produce the 

 maximum amount of work done by such a movement. 



Let ad Be any fibre, and let ab be the line traversed by 

 the point a; and draw bp perpendicular to ad. Then 



El = ad - a'b = dp = ab x cos (pab) but pad = j3 - 0 - 

 Therefore, 



mO = ab x cos Q3 -0-0) d&. 

 Let ab, which is the same for all points of the bone be called 

 A, then we have 



Work done = Ud9 = A 



cos (0-0-0) dO, 



= A { sin (/3 - <j> + 0) - sin (j3 - <f> - 9) { . 



or 



Work done = 2A cos (/3 - (f>) sin 0. (43) 



This expression is a maximum when j3 = 0, or when the bone 

 is moved parallel to itself, in a direction parallel to OX, the 

 bisector of the vertical angle of the triangle formed by pro- 

 ducing A A' and BB the two extreme fibres of the Quadrila- 

 teral muscle. Therefore, 



If there be given a fixed bone A'B and a perfectly free bone 

 AB lying in the same plane, forming the origin and insertion of 

 a Quadrilateral muscle ; when that muscle contracts, the foot of 

 the bisector of the vertical angle OX must move towards 0, but 

 the bone AB may have any position whatever, subject to the fore- 

 going condition. 



From the preceding investigation it appears that the mo- 

 tion of the bone AB is indeterminate, but subject to a cer- 

 tain condition, viz., that the point X of the bone shall move 

 towards t^e point 0 ; a condition which agrees with what 

 has been already stated, pp. 181, 195. In nature, however, 

 nothing is indeterminate, and conditions are always added, 

 which completely fix the actual motion in every case. One 

 of the most common additional conditions in nature is the 



