MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
275 
Whence determining - v x and v 2 , 
v = 
1 ) 2 ”- 2 »~ 2 + C g (e* — l) 2 ”— 2 ‘~— a 
(91.) 
s (n — 2i — 2)0 
If we substitute this value of v in that of u, determining - the constants G\ and C 2 by 
expansion and comparison with the original series, and effect some obvious reduc- 
tions, we finally get* 
.! ( d Y=ll ( d , \2zi/ 
- — wl 2 |2(r— — 1 I 2 /2 
-{(r+l) 2 "- 2i - 2 — (r— l) 2ra - 2i '- 2 }. . . (92.) 
V= 
27 r 
71—1 
T" 
* This example is given merely for the sake of the process. The fullest development of the theory of de- 
finite multiple integrals will be found in the writings of M. Lejeune Dirichlet and Mr. Cayley. Some 
results, believed to be new, are subjoined. 
1. If u—ff..dx^dx ir .dx, l <p{a l —x i , a 2 — x v .a n — x n ) subject to the inequality +— then 
l/ * / «, 2 h 1 2< ~ 
u — 
h,r 
D(D — n) 
s°-fu= 0, 
(«•) 
where A - — h,-— — hVv VK~4—„> and si=t. This is a better form of the theorem in Ex. 2. 
duf da o 2 da n - 
2. Let u=jy..dx l dx 2 ..dx„f( A) in which A={(a,— x l )‘ 1 .. + (a n — « M ) 2 } 2 Subject to the inequality aq 2 .. +x,* 
1 
and let (a l -.. + a n -)~ =r ; it is evident that u is a function of t and r. Now 
Thus (a.) may be put in the form 
T s°-r u 
l 
_ _ - - - r£ 2 ^M = 0 ; 
D'(D' + n-2) D(D + ») 
the complete integral of which, determined by Propositions 2 and 3, is 
« =^ 2 (^ - 1 ) ( f ^- 3 ) ” ” + 2 ) • (^- 1 ) ” (^- »+ 4 ) { o + >K— 0 } - 
In the case of n= 3, we find on determining the arbitrary functions 
u =^(t^-^{<P( r + t)-<p( r -t)}, 
wherein <p(r)=J'JJ'rf(r)dr 3 . A similar analysis is applicable to the theory of the attractions of spheroids. 
3. The evaluation of the definite integral / rf^(sin x) n f(r+t cosa?) depends on a partial differential equa- 
J o 
tion similar to the above. Its value is 
«=2.4..(»-l)(<|-l)(<i- 3 )..( (; l-» + 2)(i) _ ”{/(r + 0-A--0}- 
4. The evaluation of the definite multiple integral ff..dx l ..dx„f(a i x l .. + a a x„)cp(x l ..+x n ) extending to the 
positive limits of the inequality ar,... + a?„<A, depends on a partial differential equation of the first order; the 
result is 
a” -1 / d\~ n+i 
U '(°1 — a 2X a i — « j) ••(«! — «») Jo dt ^\dt) A*'* 
A particular case of this theorem has been obtained by Mr. Leslie Ellis, by means of Fourier’s theorem. 
5. By a reverse application of the method we are able to assign the complete integral of the equation 
, „d-u 
d 7 u , d"u 
