On the Expansibility of the Multiple Integral

Ludwig Schläfli

A chapter in Gesammelte Mathematische Abhandlungen, 1953, pp 271-274 from Springer

Abstract: Abstract Assuming a system of n orthogonal coordinates x, y, ..., z, measured from an origin 0, let the equation x 2 + y 2 + ⋅ ⋅ ⋅ + z 2 = 1 $${x^2} + {y^2} + \cdot \cdot \cdot + {z^2} = 1$$ (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 Δ = ( x , y , ... , z 1 , 2 , ... , n ) $$\sqrt \Delta = \left( \begin{gathered} x,y,...,z \hfill \\ 1,2,...,n \hfill \\ \end{gathered} \right)$$ 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 x = S x λ p λ , y = S y λ p λ , ... , z = S z λ p λ , ( λ = 1 , 2 , ... , n ) $$x = S{x_\lambda }{p_\lambda },\quad y = S{y_\lambda }{p_\lambda },\quad ...,\quad z = S{z_\lambda }{p_\lambda },\quad \left( {\lambda = 1,2,...,n} \right)$$ 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.

Date: 1953
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DOI: 10.1007/978-3-0348-4117-7_10

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