184 CELESTIAL MECHANICS. I. iu. 15. 



/'=v {(x"-xy+{r-yy+(z"-. y] 



/"=v{(^''-x')«+(y"-y')«+(z"-.' )'}...; 



and if we suppose dxzz^xf:=:^x"zz . . . , 8y=8y'=Syi= ,.., 

 and gz=:8z'=gz"= . . ., we shall have g/==0, S/'z=0, 8/^' 

 =0, ...; and the distances will be invariable, accord- 

 ing to the conditions of the system. We may then infer, 

 from the equation Ozz^mS^s, (I), 



Ozz^mS—; Ozz^mS^; 0:=:^mS^. (m) 



dx by hz 



For since 8x=:8a?^= ... the quantity 'SimSd's, which is the 

 sum of the partial differences with respect to x, x,,, ., must 

 be divisible by dx; and the same is true with respect to y 

 and z. It is obvious that these equations constitute the 

 first part of the proposition. 



It will still be consistent with the conditions df =: 0, 

 ^f'zzO, . . ., to suppose z, 2!, z'\ . . ,, invariable, and to 

 make Sj^ziyS'sr, ^x'zry'S'zr, . . . ; Sy =z — xS'sr, ^/^x'S-zzr, . . . ; 

 ^-ar being- any variation at pleasure [for example, that of an 

 angle described round an axis parallel to z] : and substi- 

 tuting their values in two of the equations OzzSTWiSS'*, we 



have, since Sm*S ^r-8^=SwiS-^r-y^'S^» and SmA^ ^r- dy 



I ex ex by 



=;27»<S-y- ( — ar^-sr), adding these together, and ^divi- 



ding them by ^-sr, 0=Sm5 (y^ Xy^; [the third equa- 

 tion disappearing, because 8z is supposed to vanish, as 

 when the variation takes place in a circle described on the 

 axis parallel to ^.] Fof the sauxe reasons, we may obtain 



