104 
MR. E. P. CULYERWELL ON DISCRIMINATION OF MAXIMA AND 
{ Y« 8 y Sy > dy = [Y„ 8 y Sy*- 1 ’]! - f Sy 8y«> + Y*Sy 8y-“) <fc 
[Y^SySy""'];- 
d 2 Y ns 
VZY, 
Sy 8 ?/ 
(i-D 
Jo 
and, finally, 
+ 1 
Y 0 ,S yW-»)dx; 
{ Y w 8y 8/ = L ± { (i ^ 8/ - S y Sy + Ac.) ) 
where, with the exception of the term involving By 2 , the integral involves only 
fluxions of By, and a similar reduction can, of course, be applied to any terms of the 
same form. 
Reducing all the terms involving Sp/ in (7), we get for S 2 U an expression which we 
may write 
S 2 U = L + f (A 0 8 x y 2 + 2 2A' r , 8^ dx, 
where, in the terms under the double summation sign, B } y only appears through its 
fluxions. If we determine z x by the equation A 0 = 0, a differential equation of the 
2w th order, we have S 2 TJ depending on the limits and on terms involving Sp/ only 
through its fluxions. If this expression be transformed by writing z. 2 B 2 y for Sp/, then, 
after expanding, we get an expression which we may write 
r (n- 1) (n-J) 
8 8 U = L + 2 2 B r , B z y (r) B,y (s) dx, 
j (0) (0) 
in which the highest fluxion of Sp/ is of the (n — l) th order. 
This expression can be reduced by the same method, and we get 
S 2 U = L + ((B 0 % 2 + ( 2 !> VbV, S.p/ W B. 2 yA dx. 
Determine z 2 so that B 0 = 0 for all values of x, then S 2 U will depend on L and the 
fluxions of Sp/ : transform this by writing, in the part under the integral sign, 
Sp/ — z s Spy, and reduce in the same way, and so on till we come to the last trans¬ 
formation but one, in whicli 
S 2 U = L + f (m 0 (S„_p/) 2 + 2 2 B n _ iy v B n _^A dx, 
which, treated similarly, gives 
S 2 U = L + J (N 0 B„y 2 + N u B„y 2 ) dx ; 
and, determining z„ so that N 0 = 0, we get finally 
S 2 U = L fl- J N n (S„y) 2 dx. 
