108 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
The solution now consists in determining by trial whether d can 
be separated into n”-power numbers, all different.» 
Second method: r 
In equation (1) a may sometimes be separated by trial into 
n*-power numbers, all different. 
Third method: 
Assuming b” nearly equal to but less than S, ,, and putting r for 
the difference, we have 
Saeed a Me : y (3) 
In formula (1) p and q may be any numbers chosen at pleasure. 
In formula (2) x should be chosen equal to or greater than the 
number of powers sought, and p and q any numbers that will give 
d positive, provided p be not less than z. 
In formula (3) 6 must be greater than a. 
Formulse (1) and (2) are taken from Dr, Hart* who has treated 
the cases of squares and cubes at some length. Formula (8) is found 
to be especially serviceable if a large number of powers is sought. 
Examples.—The values of S., , for n = 2, 3, 4, and 5, respectively, 
are here set down for reference’ as follows : 
Sot ee @t) Qe+)); 
x, 
S ,=42(@+); 
S,4= 30% (62*+ 152°+ 102°—1); 
8, p= te? w+ 1)? (22*+ 2a —1). 
22 
Ex. 1. Using formula (2) assume «= 5, p= 6, and q=1. 
Then d= 42, which by trial is found equal to 
1?+ 47+ 5’, whence 
?+3?+6=7*. 
*See The Mathematical Magazine. Edited and published by Artemas 
Martin. 4°. Erie, Pa. 1882-1884. Vol.1, No.1 seta 1882], pp. 8-9, 
and No. 11 [July, 1884], pp. 173-176. 
