On the Multiple Integral ∫ d n x d y ... d z $$\int {{}^ndx\;dy...dz} $$ whose Limits are p 1 = a 1 x + b 1 y + ··· + h 1 z > 0, p 2 > 0, …, p n > 0, and x 2 + y 2 + ··· + z 2

Limited merely by the last inequality, this integral (exhibiting for n = 2, 3 the area of a circle or the volume of a sphere with the radius 1) was, long ago, ascertained to be π n/2/Г[(n/2) + 1]1). But I do not know that the similar generalization of the integral representing a sector of a circle or a spherical sector on a triangular base has hitherto treated of. For this purpose, there must besides be given n limits of the linear and homogeneous form p > 0. For if the number of such limits were less than n, the integral can quite easily be reduced to a number of integrations equal to that of the linear limits given; and if there were more than n such limits, the integral may be resolved into several others having each only n such limits. We shall therefore here confine our attention principally to the integral with n linear limits.