PROF. K. PEARSON" ON THE MATHEMATICAL THEORY" OF EVOLUTION. 261 
We have now defined the chief factors which will be dealt with in the present 
memoir, and shown how they are to be quantitatively measured. We shall now pro¬ 
ceed to their mathematical analysis on the fundamental assumption that the variations 
with which we are about to deal obey the normal law of frequency. 
(3.) Correlation with special reference to the Problem of Heredity. 
(a.) Historical. —The fundamental theorems of correlation were for the first time 
and almost exhaustively discussed by Bravais (‘ Analyse mathematique sur les pro¬ 
bability des erreurs de situation d’un point.’ Memoires par divers Savans, T. IX.. Paris, 
1846, pp. 255-332) nearly half a century ago. He deals completely with the correlation 
of two and three variables. Forty years later Mr. J. D. Hamilton Dickson (‘Proc. 
Pv-oy. Soc.,’ 1886, p. 63) dealt with a special problem proposed to him by Mr. Galton, and 
reached on a somewhat narrow basis # some of Bravais’ results for correlation of two 
variables. Mr. Galton at the same time introduced an improved notation which may be 
summed up in the ‘ Galton function ’ or coefficient of correlation. This indeed appears 
in Bravais’ work, but a single symbol is not used for it. It will be found of great value 
in the present discussion. In 1892 Professor Edgeworth, also unconscious of Bravais’ 
memoir, dealt in a paper on ‘Correlated Averages’ with correlation for three variables 
(‘Phil. Mag.’ vol. 34, 1892, pp. 194-204.) He obtained results identical with Bravais’, 
although expressed in terms of ‘ Galton’s functions.’ He indicates also how the 
method may be extended to higher degrees of correlation. He starts by assuming a 
general form for the frequency of any complex of n organs each of given size. This 
form has been deduced on more or less legitimate assumptions by various writers. 
Several other authors, notably Schols, de Forest and Czuber, have dealt with the 
same topic, although little of first-class importance has been added to the researches 
of Bravais. To Mr. Galton alone is due the idea of applying these results— 
usually spoken of as “ the Jaws of error in the position of a point in space ”—to the 
problem of correlation in the theory of evolution. 
The investigation of correlation which will now be given does not profess, except 
at certain stated points, to reach novel results. It endeavours, however, to reach 
the necessary fundamental formulae with a clear statement of what assumptions are 
really made , and with special reference to what seems legitimate in the case of 
heredity. 
(b.) Theory of Correlation. —Let y x , y d . . . y u be the deviations from their 
respective means of a complex of organs or measurable characteristics. These organs 
may be in the same or in different individuals, or partly belong to one and partly to 
another individual. The complex may be constituted by a natural or artificial tie 
* The coefficient of correlation was assumed to be the same for the arrays of all types, a result which 
really flows from the normal law of frequency. 
