136 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



But the work done by the cyclic forces is 

 (5) SA = 2 8 P s Sq s =2ST. 



Hence the last part of the theorem is proved. Again, in any 

 motion 



(6) sr=s s 8 5 ; + 2 s |V 



and in an isocyclic motion 



(7) ' 8 ^|%- . ; , 

 But since the work of the positional forces is 



(8) $A = 2 s P s $q s = -^~Sq s = ~ ZT, 



uQs 



the first part of the proposition is also proved. 



II. In an adiabatic motion, the cyclic velocities will in general 

 be changed. 



Then they change in such a way that the positional forces 

 caused by the change of cyclic velocities oppose the motion, that 

 is, do a positive amount of work. For since for any positional 

 force 



P d l 



"0?,' 

 the change due to the motion is 



dST ^ <?T VT 



t>*s ~ "^ = Z r x cq r 2, r x ^JT- oq r . 

 dq 8 dq s dq r J dq s c)q r J 



Of this the part due to the change in the cyclic velocities is 



<> T-, <o &T ?- / v 3p r ^_ , 



8 *-' p '=- x 3p;^ = -^^" 



and the work done by these forces is 



B ? A = t,^P s &q s = - 2,5^1 Sq,Sq r ' 

 Now we have for any motion 



cv_ ^ 3p r cs , v dpr cv- / 



*^jgfc+s.jg&'. 



and in an adiabatic motion this is zero, so that 



