153 



xjc in the following- manner, proposed by Prof. J. C. Kapteyn \\ 

 The correlation-coefficient rj/c of Xj and xjc is that part of the 



square of the mean error common to Xj and Xk which is due to the 



common causes. 



Supposing every quantity tii to have the same mean error, or 



we find for ?]jk 



rjk = 



^ (ijl au 



Now ^ ctji'' apparently equals the number Nj of the causes acting 

 upon Xj, .2 aj-i' the number Mk of the causes influencing Xf^ and 

 ^ (tjiccki tlie number Njjc of the causes contributing both to .r^ and .77,.. 



Thus, in the case of equal mean errors, we have 



in other words : for f ^ = g^ = • • • = f^ the correlation-coefficient 

 equals the quotient of the immber of common causes, divided by 

 the geometrical mean of the numbers of the causes, which act upon 

 Xj and Xk resp. 



If both Xj and xu are subjected to an equal number (iV^=^V),.:=^V) 

 of causes, Nji of which act both upon Xj and .i^-, then 



m other words : the correlation-coefticient is that part of the causes 

 of Xj (resp. X]^ which also contributes to x^ (resp. Xj). 



The expressions for the correlation-coefficients admit of a very simple 

 geometrical illustration. 



Calling spherical simplex Sp a (^-dimensional) (5-gon lypng on a 

 (^-dimensional hypersphere (extension of the spherical triangle in 

 3-dimensional space) we may state that a spherical simplex S^ has 



Q vertices P„ P^, . . . , Pp and — ^;^ edges pjk = PjPk- 



Opposite to the vertex Pi we find, in the (9 — 'i)-dimensional linear 

 space JTj, the (curved) (c — 2)-dimensional face of Sp, which contains 

 the remaining q — 1 vertices Pj {j =\= i). 



Further we denote by jtjJc the angle between the linear spaces 

 :ntj and jtj. [consequently also between the {q — 2) -dimensional faces 



^) J. G. Kapteyn. Definition of the correlation-coefficient; Monthly Notices of 

 R. A. S., vol. 72 (1912), p. 518. 



