RELECTIOX OX THE VAPHABILITY AND COERELATION OF ORGANS. 
3 
In particular, if a selected group be not given with very great accuracy by a 
normal frecpiency surface, still we may, I think, consider ourselves justified in 
supposing that the effects of the actual selection, and those of a normal selection 
with the same mearis, the same amounts of variation, and with correlations of the 
same intensity will be at least cpialitatively alike in character, if they be not indeed 
exactly the same quantitatively within the limits fixed by the probable errors of the 
constants. 
My plan in this memoir will be as follows :—I shall first state the fundamental 
theorem in multiple correlation with a new proof, so that the formula; required may 
be once for all collected for reference."^ I shall then give the algebraic investigation 
of the new formulae for selection. I shall afterwards consider special simj^le cases, 
and illustrate them by examples. Finally I shall draw attention to the nature of 
the selective death-rate as indicated in cases of this kind, and consider at length its 
algebraic theory. Throughout I shall endeavour to illustrate the somewhat complex 
algebra by arithmetical examples. 
(2.) On the Fundamental Theorem in Multiple Correlation. 
I have shown in my memoir on “ Regression, Heredity, and Panmixia” (‘Phil. 
Trans.,’ A, vol. 187, p. 261) that if the n variables of a complex be functions of ni 
[m > n) independent variables witlr fretpiency distributions following the normal law, 
and such that the principle of superposition holds for the deviations from the means 
supposed small ; then tlie frequency of the complex with deviations from the 
means of the n variables lying between and x^ + Xo and Xo + . . . .r,, 
and Xn + Sx„ will be 2 8.ro . . . 8x„ where 2 : 
^ ^0 ^ ippq'^p^Q) ). 
Here Zq, Cpp, c^g . . . are constants, and denotes a summation for every value of 'p, 
and So for every j^air of values of p) and q in the series from 1 to n. 
In the same memoir (p. 302) I have determined the values of 2 q, Cpg in terms of 
the correlations and the standard deviations o-p and o-g of the n variables. This 
point had already been considered by Professor Edgeworth (‘ Phil. Mag.,’ vol. 34, 
p. 201, 1892), and some further residts by Mr. A. Black, reached before his death in 
1893, were published in the ‘ Camb. Phil Trams.’ (vol 16, p. 219, 1897). The 
})resent investigation is, I think, novel, and adds to results already reached others 
required in the j^i’esent memoir, so that it thus 2 )laces together with a fairly simple 
2 :)roof all the fundamental results to ’which I shall have occasion to appeal later. 
* Y e have used these formulae for several years, but they do uot appear to have been hitherto published 
in a collected form. 
