406 
MR. W. D. RIVER OR CERTAIR DEFIRITE IRTEGRALS 
Tliis can be integrated if p be a function of 9. 
In the particular case where p is constant and tbe mass is denoted by M, we find 
M (l + ^V= + 
8 v 3i \1 
2i + 3 21+1! ' ’ Jr 
• (36) 
§ 19. The result just formed exlfibits in a very simple ma nn er Maclaurin’s 
Theorem, that the attractions of confocal ellipsoids at external points are proportional 
to their masses. For, as has been already pointed out, the operator V 2 is unaltered 
by the addition of h (v-+ V+T7 
J \df dg dh 
The series (35) also shows in what direction Maclaurin’s Theorem may be 
generalised. 
(i.) Let p=f(0). We then see that if there be two ellipsoids which would be 
confocal if they were coaxal, and if the matter in them were arranged in layers 
similar to their bounding surfaces according to any specified law depending on the 
parameter of the layer, then the attraction of such ellipsoids at external points whose 
coordinates referred to the axes of the ellipsoids are equal, are proportional to the 
masses of the ellipsoids. 
(ii.) If we multiply the value of a given in (33) by 
27r v / (cr—</>)(6 2 —<£) (c 3 — 4) F (<j>) d<f> 
and integrate between the values rfp and </> 2 , we find on the right hand side 
r<#’i 
477 j x/{cr—— (f>)(c 2 — <f>)F((f>)d<f>x ( 1 + .^ 7 + 
9.1 
i + 1! 
in which the operator is independent of <£. 
On the left hand side we have 
277 y/ (a 2 — <fi) (b' 2 — cj)) ( c' 2 
d(f> 
where p is the perpendicular from the centre on the tangent plane at the point x, y, z 
to the confocal, passing through the point of parameter (f>. 
Hence, if the matter between two confocal ellipsoids be affected with a density 
varying as F(cthe equipotential surfaces will be confocal ellipsoids, as may be seen 
from the expression (37), which also gives the ratios of the attractions due to different 
shells. 
(iii.) The theorem is true for any value of V satisfying Laplace’s equation. 
