214 Prof. E\ Y. Edgeworth on the 



analogy of the case of a single variable (Philosophical Maga- 

 zine, 1896, xli. p. 95 et seqq.), and then subjecting the general 



expression to the conditions that the quantities ys &c. are small. 

 The form of the solution in series is such that the third dif- 

 ferentials -~ , 2 3 ", belong to the same form, which accounts 



for the circumstance that the functions 1 6 2 &c, satisfy equa- 

 tions (1) and (2). 



Such being the solution for principal axes, the solution for 

 any axes x ! , y 1 is found by substituting in the above expression 

 for x, x' cos 6— y' sin 0, and, for y, —x f sin 6— y' cos 0; where 



tan 2*=* '""' 



&', l ; , m! corresponding to our original k, l 9 m* ; and by 

 substituting in (10) for k, m, n, p, q, r the equivalents of 

 those coefficients in terms of k r , V, m! , n 1 , p f , q f , r f f. 



It may be observed that to whatever axes the surface be 

 referred, if we integrate between extreme limits with respect 

 to y, the resulting curve in x is a probability-curve (of the 

 asymmetric kind) J. This theorem may be employed to test 



* See p. 208. f See note to p. 208. 



X This proposition may be deduced a priori from the reasoning em- 

 ployed on a former occasion to prove the symmetrical compound law of 

 error (Phil. Mag. 1892, vol. xxxiv. p. 522). The proposition maybe verified 

 by integrating (10) with respect to y, between extreme limits ; and observ- 

 ing that, of the five terms within the brackets, the first two remain unaltered 

 because 



1 -11 



V2mn y 



i 



the third and fifth terms vanish, because 



1 



e zm xydy=0; 



and the third term vanishes because 



1 -£. 

 ==e z™xy 2 dy=m. 



I 



Thus the integration with respect to y results in an asymmetrical proba- 

 bility-curve identical with that which has been given in the preceding 

 article (z being substituted for y). 



