ERROR TO CASES OF NORilAL DISTRlBUTiOX AR-]) CORRELATION. 
103 
in these cases that not only are the curves of frequency of the separate attributes 
approximately normal curves, but the frequencies of joint occurrence of different 
measures of these attributes follow (approximately) a simple law, corresponding to 
the law of correlation of errors of observation. 
Since we can never observe more than a finite number of individuals, it is 
impossible to decide with absolute certainty as to the existence, in any particular 
case, of this (or any other) law of distribution or correlation. But if the number 
of observed individuals is large, and if they are obtained by random selection from 
a “ community ” coiuprising (practically) an indefinitely great number of individuals, 
the theory of error provides us witlr a test for deciding whether any particular 
law, suggested by the given observations, may be regarded as holding for the 
original community. 
The main object of the present memoir is to obtain formulae for testing the 
existence, in any particular case, of the normal cUstrihution and normal correlation 
described above. As the treatment of multiple correlation presents some difficulty, 
I have restricted myself to the cases of one attribute, supposed to be normally 
distributed, and of two attributes, supposed to be normally correlated. Where 
the hypothesis of normal distribution or of normal correlation may be regarded 
as established, there are different methods of treating the statistical data; and these 
may lead to different results. I have therefore given formulae for comparing the 
relative accuracy of different methods of calculating the frequency-constants which 
are required. 
The application of the formulae to actual cases is postponed until certain tables are 
completed. In the absence of these tables, Kramp’s and Encke’s tables (printed at 
the end of De Morgan’s article on the “Theory of Probabilities” in the ‘Encyclo¬ 
paedia Metropolitana’) may be used for cases of a single attribute. For cases of 
correlated attributes, I have given two methods of making a rough calculation of the 
“ theoretical ” distribution, for comparison with the “ observed ” distribution. These 
methods depend on theorems which can be conveniently expressed in a geometrical 
form. As the normal curve lends itself to geometrical treattnent, and as the funda¬ 
mental formulae in the theory of error can be obtained by the use of ordinary algebra, 
T have attempted to make the memoir complete in itself by starting with a simple 
definition of the normal curve, and adopting Galton’s definition of normal correla¬ 
tion ; and by deducing the necessary theorems without the direct use of the 
differential or integral calculus. 
The normal curve may be defined in various ways, e.g. ;— 
(1.) Fimctional Equation, z — f [x’), where f {E) X f {y^^") — f {x“y~). 
(2.) Ordinary Cartesian Equation, z cc 
(3.) Differenticd EquoAion, or {dzjdx) -f xz = 0. 
(4.) Geometriccd Equation, abscissa X sub-tangent = constant This follows at 
