208 Prof. F. Y. Edgeworth on the 



sum (or more generally an expansible function *) of a number 

 of elements £i f 2 , &c, each of which, being a function of two 

 variables x and y, assumes any particular system of values 

 according to any law of frequency £ =/*(#>#) ; the functions 

 / being in general different for different elements. If each 

 of these functions is referred to its centre of gravity at origin, 

 and expanded in powers of x and y, it appears, by parity of 

 reasoning with that employed in the case of the simple law, 

 that for a first approximation we need take account only of 

 terms of the second order. Integrate between extreme limits 

 of x*f(xy)dx dy for each element ; and let the sum of all 

 these integrals be h. Also let 



l=t^xyf i {xy)dxdy, 



the integration extending between the extreme limits of each 

 element, and the summation over all the elements. Then 

 s, the sought function which is to express the frequency of 

 Q, will be of the form 



z=®(x,y; A,-i,m)t. 



This expression may be simplified by transforming the 

 axes to new ones making an angle 6 with the old ones, such 

 that the new I vanishes. This will be effected if we put 

 tan20 = 2Z-r-(&— m)%. Thus we may write with sufficient 

 generality : — 



z=<&(x,y ; Is,m), 



By superposing a new element after the analogy of the 



* Cf. Phil. Mag. 1892, xxxiv. p. 431 et seq. 



t I use a semicolon to separate the variables (x and y) from the con- 

 stants (k, I, m). 



X Put x =X cos 6 - Y sin 6, 



*/=Xsin<9+Ycos0. 



The new I = SjJX Yf.dX dY ; 



where f x is what fi(xy) becomes when for x and y are substituted their 

 values in X and Y ; each element is integrated between extreme limits, 

 and all the integrals are summed. Transforming back to the old axes 

 we have for the new I 



Sjj/K^) (* (3/ 2 -^ 2 ) sin 20 ± W cos 20)4r dy= i(m~Jc) sin 26+1 cos 26 ; 

 which becomes null when 



tan20=2/-r(A-m). 



