582 
NOTES AND REFERENCES. 
the eccentricities of the two principal sections through the major semiaxis A, 
Lagrange remarks that, starting from this result and making use of a theorem of his 
own in the Berlin Memoirs 1792—93, he was able to construct the series by means 
of the development of the radical 1 4- Jx 2 + y 2 + z 2 — 2by — 2cz + b 2 + c 2 in powers of b, c, 
preserving therein only the even powers of b and c, and transforming a term such 
as Hb 2l>n c m into a determinate numerical multiple of | it ABC. H (B- — A 2 ) m (C 2 — A 2 ) n . 
It occurred to me that Lagrange’s series must needs be a series 
reducible to his form as a function of B 2 — A 2 , C 2 — A 2 , in virtue of the equation 
(d 2 d 2 d 2 \ 
\da? db 2 dc 2 J ^ ^ a ’ = ^ satisfied by the function (p (I wrote this out some time 
before the Senate House Examination 1842, in an examination paper for my tutor, 
Mr Hopkins): and I was thus led to consider how the series in question could be 
transformed so as to identify it with the known expression for the attraction as a 
single definite integral. 
I remark that my formulse relate to the case of n variables: as regards ellip 
soids the number of variables is of course = 3: in the earlier solutions of the 
problem of the attraction of ellipsoids there is no ready method of making the 
extension from 3 to n. The case of n variables had however been considered in a 
most able manner by Green in his Memoir “ On the determination of the exterior 
and interior attractions of Ellipsoids of variable densities,” Carnb. Phil. Trans, vol. v. 
1835, pp. 395—430 (and Mathematical Papers, 8vo London, 1871, pp. 187—222); and in 
the Memoir by Lejeune-Dirichlet, “ Sur une nouvelle méthode pour la determination 
des intégrales multiples,” Liouv. t. iv. (1839), pp. 164—168, although the case actually 
treated is that of three variables, the method can be at once extended to the case 
of any number of variables: it is to be noticed also that the methods of Green and 
Lejeune-Dirichlet are each applicable to the case of an integral involving an integer 
or fractional negative power of the distance. This is far more general than my formulae, 
for in them the negative exponent for the squared distance is = \n, and, by differen 
tiation in regard to the coordinates a, b,... of the attracted point, we can only change 
this into \n + p, where p is a positive integer. But in 28, the radical contained in 
1 
the multiple integral is 
where s is integer or fractional, and by a 
{(«! — x x t) 2 -(- 
like process of expansion and summation I obtain a result depending on a single integral 
J + ^ p-. And in 29, retaining throughout the general function cp (a x — x 1} ...) 
and making the analogous transformation of the multiple integral itself, I express the 
integral 
in terms of an 
V = jdx 1 ... dx n ¿r 1 2ixi+1 ... #/ +1 2l v+i... cp («j — x x ,...) 
integral í T n+1 (1 — T' 2 ) k+f WdT, where 
0