1898.] of a certain determinant, etc. 3 



I was asked some time ago by Professor Karl Pearson to 

 evaluate J for the case of n = 4, as this determinant had presented 

 itself in his " Mathematical Contributions to the Theory of Evo- 

 lution*." 



In the case of n = 3, which I tried first, the Jacobian was 

 evaluated without difficulty, by straightforward algebra. It was 

 pointed out to me by Mr L. Crawford, Fellow of King's College, 

 to whom I shewed the problem, that in the case of n = 3, R was 

 the equivalent of a familiar determinant in spherical trigonometry, 

 half the square root of which is (with the usual notation of 

 Spherical Trigonometry) the expression 



Vfsin s sin (s — a) sin (s — b) sin (s — c)}, 



sometimes called the Staudtian ; and Mr Crawford shewed that J 

 could be easily evaluated in this case by means of the formulae of 

 spherical trigonometry. 



This naturally led me to try to interpret the general case in 

 terms of the geometry of elliptic space of n — 1 dimensions, or — 

 which is the same thing — in terms of that of the surface of a 

 hypersphere in ordinary space of n dimensions. It appeared that 

 if rij were taken to be the cosine of the " distance " between the 

 vertices i and j of a polyhedron formed by n points 1, 2, ... , n, 

 in elliptic space of n — 1 dimensions, then r'y was the cosine of 

 one of the two " angles " between the two faces opposite the points 

 i and j. From the known symmetry between distances and angles 

 in elliptic space it at once followed that the relation between the 

 quantities r^ and the quantities r'% must be reciprocal, a result 

 which it is easy to verify algebraically, and which is of great 

 assistance in the work. 



In § 2 I establish the connection with elliptic geometry and 

 the reciprocal relation just referred to. In § 3 the result to be 

 proved is stated, and the Jacobian is reduced to the quotient of 

 two Jacobians J 1} J 2 of lower order. In § 4 the first of these is 

 evaluated by a process depending on the assumed value of J for 

 the case of n— 1. In the course of this work I get a rule 

 (equation (7)) for the reduction of a symmetrical determinant of 

 order n to one of order n — 1, a result which seems to me of some 

 interest if it is not already well known. In § 5 the Jacobian J 2 is 

 evaluated. In § 6 the results are collected. 



* See Pearson and Filon :"On the Probable Errors of Frequency Constants and 

 on the influence of Eandom Selection on Variation and Correlation." Phil. Trans. 

 A. vol. 191 (1898), pp. 229—311. 



1—2 



