MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
273 
If we write the above series in the form u=l,uj m , then it is easily seen that 
l 
u 
m m(m + n) 
tfUm - 2 = 0. 
Hence, if 
it- 
D(D + w)‘ 
(84.) 
" r (0 
D (D -f- n) u — X 2 z 26 u = 0 . 
Restoring t, and for X 2 writing 
d ' 2 d 2 d 2 
"i da 2 '^ 2 da 2 ^ n n d a 2 ‘ 
and then dividing the result by t 2 , we have 
d?u n + 1 du 
dt 2 
dt 
A 2.^.. 
.A 2^. 
” 2 c/a 2 
0, 
» c/a 2 5 
(85.) 
which is the equation sought. 
Mr. Green*, considering a particular case of this multiple integral connected 
with the theory of the attractions of ellipsoids, has obtained an equation different 
from the above, and not involving the constants h x h 2 ..h n . It might be worth while 
to inquire whether the equation (85.) does not more precisely define the function to 
be determined than Mr. Green’s equation does, and whether an analysis might not be 
founded upon it which should be more simple, and less dependent on foreign consi- 
derations. It would too much extend the limits of this paper to enter into the general 
discussion of the equation here, and I shall therefore merely observe that it is re- 
ducible whenever n is an odd integer, to the form 
d 2 u 
W 
7 „ d 2 u 7 0 d 2 u . „ d 2 u 
, A 2 — — A 2 ... A 2 — 0 
"i da 2 >1 ‘ 2 da 2 n da 2 —"' 
( 86 .) 
To prove this, we remark that the constant in the second member of (84.) may be 
rejected, because in operating on both members with D(D-J-n) it disappears. Writing 
then the equation in the form 
-\2 
u ~-D(D + nf u = 0 ’ 
X 2 
: 
let us assume ^~"D(D— l) 
By Propositions 2 and 3, C. § 1, we find 
u = s -»'(D — 1 ) ( D - 3) .. (D — n + 2) v 
=?( 4 - i )( 4 - s )-( 4 — +»>• 
(87.) 
Then in (87.) making t*=t, we have 
d?v 
dt 2 
d 2 v 
— —X 2 v = 0, 
which is equivalent to (86.) 
* Cambridge Philosophical Transactions, vol. v, 
2 n 2 
