Generally we have, the number of variables being n,

(4) M. Catalan* has shewn how to evaluate a definite multiple integral which depends on those which have just

2

x2

in which = 1 − x ̧2 − x2 - ... - -1, and the limiting

Liouville's Journal, Vol. VI. p. 81. The reader is referred to a paper by Mr Boole in the Cambridge Mathematical Journal, Vol. 111. p. 277, entitled 'Remarks on a Theorem of M. Catalan,' where the truth of the Theorem is called in question.

values of the n-1 independent variables are given by the

and the other coefficients are subject to the conditions,

The number of constants in (2) is n (n - 1), the number of conditions in (3) is n (n + 1), so that there are n (n-3) arbitrary coefficients.

Adding the squares of (2) we have by the conditions (3)

u 12 + u22 + + u„2 = x2 + x2 + + x2 = 1.

...

From the same equations we find

...

Also on changing the differentials by the method given in Chap. 11. Sect. 2 of the Diff. Calc. we have

If now we integrate with respect to all the variables except u, the limits being given by the condition

sin u cos v, Xz = sin u sin and if we

v,

take the limits from u = 0 to u = π, and from

= 2, it takes the form

0 to

(d) £"£2 du dv sin u ƒ (a, cos u + a, sin u cos v + a3 sin u sin v)

= 2πf" de sin 0 ƒ (A cos 0).

The formula under this shape was first given by Poisson

in the Mémoires de l'Institut, Tom. 111. p. 126.

if we put 1 = xx, the corresponding limits are

The only restriction on the generality of this result is that a must be less than r.

(b)

If r = 1, we have

= 1

This last integral may be considered as being made up of two parts, one from 0 to x = 1, the other from z to x = . This second part may be reduced to the same

In the formula (b) put ≈ = y2 and b = 2a, when we find

up of two parts, one from = 0 to x = 1, the other from a = 1

1

to x = ∞. If we put

30

for a, the latter part becomes

-fo'dx

x