210 CELESTIAL MECHANICS. 1. V. 21. 



We may obtain from the equation (P) (317) the particu- 

 lar value 0=2wi ^ — ^I^— + 2m (Py—Qx), if we cause 

 the variation ^x to disappear from the expression ^f=.^^/ 

 |(i>'— x)2-f(y— yy-H(/— z^l by making 5x'=:^ + a^<; 



y y y 



-^^y" /•> •'• '-> [t^^ P^*** of each of these expressions, 



y 



that involves J^or, belonging to a supposed revolution of the 

 body round the axis parallel to z : for if the distance of m 

 from this axis be s, and that of m\ s\ the elementary arc 



described by m will be — ^x, and the arc described by 



V 



Off* 7/ 



m', — . — ^xzz — S.r, whence the variation of j:' will be -^. 

 s y y . s 



— Sxzi^Sa;]. This substitution gives us the value of ^, 



y y 



^f'i ^f^f '" * independently of ^x, [as it must necessarily do 

 from the agreement of the variations substituted with a 

 rotatory motion] : we are therefore at liberty to assign any 

 value to ^x at pleasure, while we observe these conditions, 

 and its coefficients may be made to vanish, [as they must 

 obviously do if ^x be infinitely greater than the other varia- 

 tions concerned]. In making this substitution for dx\ . . . , 



in the equation (1*) (317), thatis,Oz=mSx (— ^ — P\ .. +, 

 m'^s' (— — — P'\ ... we are only required to employ for dx\ 

 ^ — , since Sir', is supposed to vanish in comparison with 



