Karl Pearson 
31 
Now, if you take 
X = Am + Bn + . . . , 
1/ = A'lii + B'li + 
f.r' = A" a III'- + B-aii' + . . . , 
a If- = A'-a-„i- + B'-aii' + ... , 
M ean ( hx 81/ ) = /•.)■</ o".,- o",, = ^1 ^4 'a,,,- + BB'an'- + 
Whence 
a/a^ (1 - r,/) - {A^B'^ + A'^B'^ - 2AA'BB') a,^a;,^ 
= S{AB' - A'Braii^a„r, 
whence, rcuiemberiiig //,„ = ^r—^,, /'« = , 
we easily deduce the 
Z = , e 2 V<r,,- (r,.(7„ <T,;V 1 - r^,/ 
27ro-^.o-y VI - r^./ 
of our familiar notation. 
But this is precisely what Bravais does not do, and for the simple reason that 
his X, y, z are not variables Avhich he has directly determined and for which he can 
directly find o-^., ay and r^y. Ho is merely seeking to express the variability of x 
and y in terms of the directly determined constants and certain differential co- 
efficients. This is one of the fundamental problems of the Method of Least Squares 
and had already been solved by Gauss. Bravais adds so far nothh[i whatever to 
Gauss' solution of 20 years earlier. If Bravais discovered correlation, then Gauss 
had done so previously. 
As a matter of fact while the above expression shows how a hasty examinati(_)n 
of Bravais' memoir might lead one to believe he had reached the correlation surtacc, 
he was in fact occupied with a,n entirely different problem, one which was really 
only a particular case of Gauss' earlier and more comprehensive work. 
We camiot pass over, however, the really valuable portion of Bravais' memoir. 
It lies in this : Having got his coefficients of x and y in terms of the differential 
coefficients A, B, C,'... he writes the surface 
IT 
and then discusses the properties of a surface of which the contours are 
(/,/■■- + 2cxy -\- by- = D, 
i.e. the familiar ellipses of our normal surface. He gets the conjugate of j'-axes as 
the locus of maximum //'s and determines the probability of jwints lying in certain 
areas — bounded by similar ellipses or in angular sectors. He gets the line 
x = y, which corresponds to Galton's regression-line. But this is not a result of 
observing and y and determining their association, but of the fact that x and y 
