96 Prof. F. Y. Edgeworth on the 



To determine the absolute term in the expansion of y, it 

 may be observed that in the case of symmetry fa vanishes 

 (since y cannot involve odd powers of x) ; whence it appears, 

 since / also vanishes in this case, that fa has no absolute 



A 



term, fa reduces to — --- . A is found by condition (b) to be 

 1 \'k 



/^— , while the value of y is (the expansion in powers of x of) 



the well-known (symmetrical) probability-curve. 



To proceed another step, let the first term of fa be Bi -J— 9 



a form which is prescribed by equation (1) ; and the second 

 term of fa, 



K J_ 



By equation (6) combined with (5) we have 



(«+i'(«»)+-)&+( l+ i« , (*s)+-)£ 



~ ~ §K 2 dk) + |3 xS V dV + -- )dk VIS' 

 This identity evidently requires that Ai = G. The identity is 

 then satisfied by B,= -=• 



It is unnecessary to proceed further with the determination 

 of the coefficients, since the higher powers of the expansion 

 may be neglected. We have therefore for the solution 



H 1+1 A*T>-)V7 7I 



an expression of which the expansion in powers of x proves 



to be identical with the expansion of the formula given above, 



(? 

 when h is replaced by — • 



a 



Second Method. — The following is perhaps a simpler solu- 

 tion. Put as the correction of the first approximation 



1 jt? 3 



( V1Z - l J= 7= AT; * 2 *) 



