Laplace's " Mecanique Celeste." 1 * 



a) = '-^^3 for, a) = iv'-^ =.^.'' 



(^— ) = ,; since — -jz— — — di dz — 



1 t?.sm* _ sin <r lif^iZ; _ / ^ ^ ^ *i^ X 

 "Z^' dz cos'Jo- ' dz Vds/ cos^tr 



/^^ = i . ri - sin ^ TT + COS* 9 . sin ^ ^] = 7- • [ 1 — 



r /''"''"X d . (sin ^ . sm t) __ 



sin * TT . sm ^ 6] ; for, (^-^ ) = ^^; - 



sin 9. cos ^.(^) + sin7r. cos9.(-^). 

 (i^\ = J2!i_.[cos^7r-2.sinM.sin*7r]; for, (^) 



' cos i . COS * / dv\ 



(^ ;; / cos ^. cos* / dv\ sin*.sin^ 



(dli \ cos ^. sill ^ { £L\ 



Also, 



-2.sinT.c0s* .- \ dz / _ Vr.sin^/ 



cos rr 

 r^. sin tf 



(rfJ^rx — 2. sin T. cos* - \ dz / \r. si 



-—\ =r : : since = r 

 dV' ) r-.sin'O ' di% dz 



— sin r / '^* \ cos * . cos ^ {" — \ 



r. sin i ' \^/ ~~ r. sin'^ ^ ' \ dz / 



(±^)is = 0. Hence, 



dx- ~ \ di' ) ~ \ dr'/ ' dx "^ \ dx J ' dx 



(fV 



