206 Dr. L. Isserlis on the Variation of 



lated from a sample extracted out of a normal population, 

 that 



V«fc= (A, 



where n is the population of the sample. In the present 

 paper we shall first obtain a series of formulae giving the 

 mutual correlations of the deviations from their mean values 

 of the various multiple, partial and ordinary correlation 

 coefficients in the case of three variables. We shall then 

 show that formula (A) is true when we may treat the 

 variations that occur in the values of the primary statistical 

 constants as differentials. But a little consideration will 

 show that the probable error of R123 as obtained on this 

 assumption can have only a restricted validity. The multiple 

 correlation coefficient is essentially positive. Its own fre- 

 quency distribution must therefore be unsymmetrical. Con- 

 sider a sampled population for which R is zero. In any 

 particular sample the value obtained will be positive or 

 zero, and hence rejecting the trivial and highly improbable 

 case in which every conceivable sample is exactly similar to 

 the sampled population, the mean value of R in many 

 samples will be positive, i. e. will differ from the true value 

 or value in the sampled population. We shall show that 

 the principal term in this excess of the mean value over the 

 true value is (1 — R 2 ) 2 /2wR, and that correct to \\n formula 

 (A) remains Itrue when allowance is made for this dis- 

 placement in the position of the mean. 



We shall in a later paper discuss the corrections to be 

 applied when n is not big enough to justify our neglecting 

 1/ti 2 , but it appears from unpublished results already 

 obtained that these corrections are very complicated. At 

 the same time the above formulae are very accurate when 

 n is not small and R is not nearly zero. Thus the cases 

 covered by the present paper include the majority of those 

 that occur in practice. 



§ 2. (i.) To express the correlation coefficients r 12 , r 13 , r 23 

 in terms of the partial correlation coefficients l r 2 z, 2^31? 3^2* 

 We have 



^12—^13^23 ,-,x 



•""-•(l-i^tt-ri,)' • • • * (1> 



and two similar equations. Consider a spherical triangle 

 ABC in which cosa = r 23 , cos& = r 13 , cosc = r 12 . It follows 

 from equations like (1) that cosA = 1 r 23 , cosJ$ = 2 r llt 



