298 CELESTIAL MECHANICS. I. vH. 35, 



It is obvious that the coordinate x will be, at the end of 

 the time t, {r-{-as) cos (5 + aw); and the projection of the 

 radius, on the plane perpendicular to x, being (r+as) sin 

 {9-j-au) we shall have 



y = (r + as) sin {6 + au) cos (wf + -sr + av), 



z=i(r + as) sin (5 + aw) sin (wf + w-fau); and substituting 



these values in the equation (F) (363), that is SF — — = 



? 

 _ dda: « ddy . dd^ i .. ^i o 



'dF'^^l^'^ 'dF'' ^^S^^^^^^S *^® square of a, 



[and calling x, ^ cos fJt>; y, v cos ^ ; and z, v sin Q, we have 



pxiz^A. cos /(A— 3)ot.A sin /A 



d^ird^A. cos /[* — d-/*x sin /t«, since dx d/*=0, and d/t^nO, 



these quantities being multiplied by a^ . 

 gxd-x=^x(d"A cos V— dV?^ sin cos /*) — ^fjt(d^x,\ sin cos /tc— 



dV^ sin 2/*) 

 Jyzz^y. cos I — SI . V sin I 



d^i^izd^v. cos I — 2dvd|.sin |— d^l.ysin |— dl^.v cos | 

 ^d^yziJ^^d^y. cos 2|— 2dyd|. sin cos |— d^l.vsin cos I— d|*. 



vcos^l) 

 —SI (d^y.v sin cos |— 2dyd|.v sin^l-^d^l.v* sin «|— d|«K« 



sin cos I) 

 ^zzzh. sin I + SI. V cos | 



d^znd^v. sin |+2dvd|. cos |-f d^l.vcos |— d| v sin | 

 ^zd2z=8<d2v.sin2| + 2dvd^. sin cos ^ + d^^.v sin cos ^—d^U 



sin 24) 

 + S^ (d^v.vsin cos^-i-2dvd|.vcos2^+d«|.v2 cos«^ — 



dl^.v^sin cos I) 

 gyd2j^4-S2d22.=8v(d2.^d|*.t)^SK2dvd|.i/4-d2|.i^ 



