AND THE FIGURE OF THE EARTH. 
// 
the limits of integration being 1 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. 
q d x dy d z 
Then v =fffiT7Z 
t ; and, by differentiating 1 V, we obtain the 
+ (g-yY + ( h—zf }* 
d 2 Y d 2 Y d 2 Y 
well-known property -jp + -jyz + YTW- ~ 0, or — 4 r according as the attracted 
particle is not or is within the attracting mass ; §' being in the latter case the density 
of the attracted particle*. By transforming these equations to polar coordinates, we 
obtain 
(PV , 2 dY 1 d 2 Y , . dY , 1 d?Y n 
+ + + r cot0 d9+^sitf9~df- o > ov-Ak?, 
and 
q r f2 d r 1 sin 9' d9' d <p' 
r'r r 7r i r 
0 0 0 i 
{r 2 +r 12 — 2 r r' (cos 9 cos 6 1 + sin 9 sin 9' cos (<p — <p')) }* 
where r 2 = f 2 + g 2 + h 2 , cos 6 = ^ + tan <p = y ; and similar expressions 
are true of r 1 & and <p' in terms of x, y, z. 
Put cos 0 = (a, and cos O' = y! , and they become 
d 2 (rY) 
dr 2 
+ 
d y 
y r r r l r 
v J 0 J -\Jo 
(( 1 -*>S)+T^ 
d 2 Y 
a* = 0, or — r 2 '!'. 
2 7T 
ft? d <p 
q r 12 d r 1 d fjJ d <p* 
{r 2 + r' 2 — 2 r r 1 (y y' + \/(l — y 2 ) \/{l — /x' 2 ) cos (<p — <p')) }t 
3. Expansion by the binomial theorem shows that 
{r 2 _|_ r '2 _ 2 r r' Qjj yJ + -/(! — ^m, 2 ) v' (1 — y/ 2 ) cos (<p — £>'))}£ 
may be expressed either in powers of r or of r ' ; thus 
( 1 .) 
( 2 .) 
P 0 vr + 1/2 + • • • + P m Vi 
+ or P 0 i + P,-'i- r 
yz 
r 1 * 
yl I 1 yz I * * * ~ I - A n j , J n -{- 1 ~ 1 " * 5 wl 0 f ~~ T - 1 1 j.2 I P 2 y ~i~" • ‘ H l - P^ H f~ • • •) 
where P re is a symmetrical function of y, V^l — yJ 2 ) cos <p, — y 2 ) sin <p on the one 
part, and — y’ 2 ) cos <p',V(\ — yJ 2 ) sin <p' 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 
r r r i r 2 * m p„\ , 1 s?v n , ,,„„■) 
JoJ-iJo fjrfjxU 1 ^ ) dy) + 1 - /X 2 df + n ( n + } ) P» j r ln - 1 
= 0, or — 
except when n — 2 ; and in all cases 
j-f. ((* — !d) t£) + f^? 1 ^p , + «( H + 1 ) p . = °» (3-) 
which is the equation of Laplace’s coefficients. 
* Vide Pratt. Mec. Phil., § 168. Laplace, Mec. Cel. liv. iii. + Vide Pratt. Mec. Phil., § 169. 
