2 Mr Berry, On the evaluation [Oct. 31, 



Ordinary Members of the Council. 



Mr A. Scott. 



Mr J. E. Marr. 



Prof. H. Marshall Ward. 



Mr A. Harker. 



Mr A. Hutchinson. 



Prof. G. D. Liveing. 



Mr S. Skinner. 



Mr H. Gadow. 



Mr D. Sharp. 



Prof. J. J. Thomson. 



Mr L. R. Wilberforce. 



Mr A. Berry. 



The following were elected Associates, having been nominated 

 by the Council : Mr R. S. Whipple, Prof. J. C. M c Lennan, Mr 

 R. S. Willows. 



The names of the Benefactors were read. 



The following Communications were made to the Society: — 



(1) On the evaluation of a certain determinant which occurs 

 in the mathematical theory of statistics and in that of elliptic 

 geometry of any number of dimensions. By Arthur Berry, M.A., 

 King's College. 



§ 1. Introduction and Summary. 



Let R denote the symmetrical determinant of order n of 

 which the constituent of the ith row and jth column is r$j, where 

 fij =rji and ru— 1, so that there are \n (n — 1) distinct quantities 

 r ij} which are to be regarded as independent variables. Let R tj 

 denote the minor* of r#, and let Ri denote that of ra. Let 



r{j = liij j *J J%i-tij . 



Then it is required to evaluate the Jacobian 



j- _ o (r i2, r is, ... , y n-i, n) /-i \ 



v 12 > ^13 > '•■ > Vn—\ , n) 



the suffixes being taken (for convenience) in the order : — 



12; 13, 23; 14, 24, 34; ...; In, 2n , ..., (n-l)n. 



* I use " minor " (in the sense which seems almost always the most convenient) 

 to denote the result of erasing a row and column and affixing a sign according to 

 the usual rule. This agrees with the notation used by Weber and Gordan and with 

 the usage of Salmon, though according to his formal definition and the usage oi 

 several common English text-books no sign is to be affixed. 



