140 CELESTIAL MECHANICS. I. ii. 12. 



— , and q—~, we shall have ^=p —, and -r- = p — - 

 dw ^ dzz dt ^ dt dt^ dt^ 



. , du ddu . du^ , , 



^ dt^^^ dF ~dF ' equation (i) will be- 



^ ddw , du^ , dz dw „ d^^2 



comeO=^— + ? — + <,_ + .;, _+„p. _ or. d.- 



expressed, in the original notation, by 0= — ^ +wi _f. + 



d^- ;j7,7 + d7(;^V)'"^^* In this equation we may 



dw^ 

 destroy the coefficient of ■—, by making gr + np^—O, that 



Q t 



is, ■~ + np^-zO, and -4- + dM=0, whence — rzw + c, — = 

 du nf- np p 



du 



n{u 4- c) and pdui= = d*, consequently «= - hl(M + c) 



n{u + c) ^ -^ « ^ ^ 



4-c', or z: hi j h(u-{-c) ?^, k and c being constant quan- 



tities : and supposing u to begin with s, we have Ac^zzl ; 



and it will be simplest to make A=l, and c= I, so that s 



X 

 becomes =i hl(M+l)", ns=:h](u-\-l), and M+l=:e"^ if hi 



e=:l, whence u — e"^ — 1, and »= -r-- = = j; e-«*. 



We thus reduce the equation to Ozz-r — H-?w-^+.^— 



^ d^2 d^ J^^dj^; 



and supposing u to be small, the last term is capable of 

 being developed in the form of a series ascending accord- 

 ing to its powers, which will be of this form, ku-\-M-\-,.,, 

 i being greater than unity, so that the equation will be- 



come Ozz-^— -^ m --- -{■ ku -\- lu* -^ . . . In order to obtain 

 dt^ dt 



