ON A MULTIPLE INTEGRAL 
[44 
power, it is worth while to remark that, by first transforming the |n th power into an 
exponential, and then reducing as above (thus avoiding the general differentiation), we 
should have obtained 
U = 
„ N ^ ri eld ds d^- 1 e~° < S +J +2W > s-*-« (s + e~ 
r (i~?) r (2^ + ^) r (Ì^-^) io Jo 
which reduces itself to the equation (14) by simply performing the integration with 
respect to 6; thus establishing the formula beyond doubt \ The integral may evidently 
be effected in finite terms when either q or q — ^ is integral. Thus for instance in the 
simplest case of all, or when q = — 
Trim—d 1 r - dxdy... 
U== vT$(n + l) (j + 2 uv)* {n ~v ~ J- 00 (x 2 + y 2 ...+ v 2 )* (w+1) {(x - a) 2 + ... u 2 }^ n ~v 5 
a formula of which several demonstrations have already been given in the Journal. 
The following is a demonstration, though an indirect one, of the formula (11): in 
the first place 
j' X {V(s + 4uv) + V+ {V(g + 4uv) — e _ 6s 
Jo \Zs\Z(s + 4<uv) 
= 2r (f~ g ) 6 " e2UV - r (4u 2 v 2 + xy-ie i6 *dx (16), 
Vtt (4mv) 2 3 Jo 
(where as usual i = \J — 1) : to prove this, we have 
j °° (4mV + x 2 )i~ì e iex dx = YjTZg) / dx / dt e ~' '*** + * ) ^ 
Jar 
r (W) 
di r 1 -® ér 4wVi -s ; 
or, putting 4wr \]t — J(s + 4?tv) ± V« (which is a transformation already employed in the 
present paper), the formula required follows immediately. 
Now, by a formula due to M. Catalan, but first rigorously demonstrated by M. 
Serret, 
cos ax dx or 
f q- (a+2z) (g _j_ a )«-i z n 1 dz, 
J 0 
'o (l+x 2 ) n (IV) 2 
(Liouville, t. vili. [1843] p. 1), and by a slight modification in the form of this equation 
j (4u 2 v 2 + x 2 )^ e iex dx = ^ ^ j s-«-* (s + 4uv)-®-* e~ 0H ds, 
which, compared with (16), gives the required equation. 
1 A paper by M. Schlômilch “Note sur la variation des constantes arbitraires d’une Integrale definie,” 
Creile, t. xxxiii. [1846], pp. 268—280, will be found to contain formulée analogous to some of the preceding ones.