On the Expansibility of the Multiple Integral

Assuming a system of n orthogonal coordinates x, y, ..., z, measured from an origin 0, let the equation (being that of a n-sphere having the origin for its centre) be satisfied by n points 1, 2, ..., n, that is to say, by n sets of coordinates (x ₁, y ₂, ..., z ₁, ..., (x _n , y _n , ..., z _n ), and suppose them so arranged that their determinant is positive (I disregard the case where the determinant vanishes). Let the signs of summation ∑, S refer respectively to the coordinates x, y, ..., z, and to the points 1, 2, ..., n, and put ∑(x ₁ - x ₂)² = u ₁₂, ... (so that u ₁₂, etc. denote the n (n - 1)/2 squares of distances of the n given points on the n-sphere); u ₁₁ is to be understood to be = 0. Determine the n linear and homogeneous functions p ₁, p ₂ > ..., p _n of x, y, ..., z, by the n equations so that p ₁, p ₂, ..., p _n may be regarded as oblique coordinates of the point P whose orthogonal ones are x, y, ..., z.