AND THE FIGURE OP THE EARTH. 77 



the limits of integration being determined by the equation to the surface of the 

 body. 



2. Let V represent the sum of the products of each particle by the reciprocal of its 

 distance from the attracted point. 



Then V =JjJ^ {f-xf +\g^'-y)^ I {h-z)^}i ' ^"^' ^^ differentiating V, we obtain the 



well-known property "^ + "^ + -^ = 0, or - 4 r ^', according as the attracted 



particle is not or is within the attracting mass ; f' being in the latter case the density 

 of the attracted particle*. By transforming these equations to polar coordinates, we 

 obtain 



dr^ '^ r dr^ r^ def^ "^^ ^^^ ^ dO "^ W^^OTf 

 and 



y _ /*»• /^'^ /*2 ,r Qr^dr^sme'd0'd(f>' 



{r8+;J* _ 2rr'(co8^co8^+ 8in ^ sin ^' cos (f - ^0))*' 

 where r^ =/2 + ^2 ^ h"^, cos 6 = ^^y-e _^ ^2 ^ ^2) , tan ^ = ^ ; and similar expressions 



are true of r' & and <p^ in terms oi x, y, z. 



Put cos & -=■ ^, and cos (f = [h', and they become 



V— r"" r^ Z*^" qr^^dr^ditJd^ 



— JqJ-iJq {r« + 7^2- 2 rr' (]*|x' + -v/(1 -f«*) -/(!--/*'') cos (f-^))}i ' ^^'^ 



3. Expansion by the binomial theorem shows that 



{y.2 + ^2 _ 2 r / (|a,|a,' + ^(l - IL^) ^ (1 - /U<'2) COS {<p - p'))}i 



may be expressed either in powers of r or of / ; thus 



Po 7 + P,;^ + -. + P,;7Sn + ••><"• Pot + PiF + P2? + -- + P.;=Ti + ---. 



where P^ is a symmetrical function of ;«/, \/{\ — yiJ*-) cos <p, \^{\ — ju.2) sin ^ on the one 

 part, and ^', \^{\ — yJ"^) cos (p', \/(l — /m.'2) sin ®' on the other. 



By substituting the first expansion in (2.), and the value of V so obtained in (1.), 

 we have a series of equations 



= 0, or — 4t^', 

 except when w = 2 ; and in all cases 



^(('-f'^)S) + i4;r# + «(« + ')P, = o, (3.) 



which is the equation of Laplace's coefficients. 



* Vide Pratt. Mec. Phil., § 168. Laplacb, M^c. C€\. liv. iii. t Vide Pratt. Mec. Phil., § 1G9. 



