a)i Ellipsoid at an External Point, 459 



only when r, as compared with the dimensions of the attracting 

 body, is very large — as, for instance, in the investigation of 

 the disturbance of the moon's motion produced by the non- 

 sphericity of the earth, and of the reaction of the same dis- 

 turbing force on the earth, causing lunar nutation and preces- 

 sion. But the fact is, that the first two terms of the series are 

 sufficient for any external point, however near^ provided the 

 square of the earth's ellipticitv be neglected ; for the ratio of 



the successive terms is not of the order -2, but — ^ — , a, h 



being the semidiameters of the earth. 



If P„ be Legendre's coefficient of the order n, then we have, 

 fjL being the cosine of the angle between r and p, 



(r^-2rp^ + pr'= ^ + 7^ + ^' + ^^ • • • = 



consequently 



f Q,.dm = f F.p'dm. 



Now we require only the even coefficients ; and their values 

 (Todhunter's ^ Functions of Laplace, Lame, and Bessel,' p. 4) 

 are : — 



5.7 , 3.5^ o . 1 



^"274^ ""2:4 '^ ■*'2.4' 

 T>_ 7.9.11 , 5.7,9 ^ , , 3.5.7^ , 1.3.5 

 ^«"" 2.4.6'^ "27476^^ "^ 27476 "^^ ""27476' 

 whence the values of Q, since p/jL=a'j are 



7.9.11 5.7.9 ^_,.,, 3.5.7 ,_,, 1.3.5, 

 ^= -27476 " - 27176 ^"^ P + 274r6 ^^P' 27476 ^ ' 

 For the integration of Qg we have 



and there is no difficulty in arriving at the equation 

 JQ,A»=- g {P/{el-el) + F,"iel-e^,) + P,"'ie^,-el)}, (2) 



