L. ISSERLIS 
395 
But S „„i (Xi - a'l) (2 „,*'i - -^i)} 
2.. .Ill 
'l...m [ V O-o 0-3 A (T., J) 
and N(Tia-.,')\2= S [ih n,{rCi—Xj){x2 -'X2) 
I. ..Ill 
= H \n., ,a {x., - X.2) (2, - ^l) 
2.. m 
with similar values for rj^, rss, etc. 
.•. the right hand side of (24) 
= - h,a;-i\., - - brAiO-i'r.is ■ ■ • 
- o-{^bj3i\.i - 6i:i6i,a-i-r,;! - 613- o-f - hrAiO-\ ■ ■ ■ 
= O-lby, (- 7-12 - 61., - 613 Tog - 6i,''24 - ■ • . - hm.r.2iii) 
+ (25). 
Each line in (25) is identically zero from the definition of the 6's and the properties 
of the determinant in (19). 
S {ih ,n.{'2..iJ'h- ^if] 
Hence - nM^' - — (26), 
JS a-c 
so that the fundamental properties proved by Professor Pearson in connection 
with the correlation ratio rj, hold for the generalised H defined in this section. 
In particular equation (26) shows that a necessary and sufficient condition for 
linear regression in multiple correlation of m variables is that 
2,3, ... — 2,3, . . iii-'-^i • 
For in this case the mean value of any an ay of x-^ will lie on the " best fitting " 
m dimensional plane. 
§ 4. The regression surface of z on xy being assumed of any particular type 
the constants in the equation may be deterniined (i) by the method of least 
squares, i.e. by making the sum of the squares of the deviations z^y — 0 (xy) a 
minimum, 2 = <^ (ic, ?/) being the regression surface, or (ii) by giving such values 
to the constants that the coi'relation between z and 0(ic, y) shall be a maximum. 
When </> {x, y) is of the second degree the two methods lead to identical equations 
for the determination of the coefficients. 
The same equations are also obtained if the surface be "fitted" to the means 
by the method of moments. There is, however, a distinction to be observed. The 
equation 5 = 0 {x, y) when the regression surface is of specified degree contains a 
definite number of constants and the first two methods will give exactly as many 
independent equations as there are constants to be determined. The method of 
moments will give as many equations as we please if sufficiently high moments 
