108 
PHILOSOPHICAL SOCIETY OF WASHINGTON. 
The solution now consists in determining by trial whether d can 
be separated into ?i**-power numbers, all different. 
Second method: 
In equation (1) a may sometimes be separated by trial into 
n^*-power numbers, all different. 
Third method: 
Assuming nearly equal to but less than and putting r for 
the difference, we have 
= . ( 8 ) 
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 x. 
In formula (3) h must be greater than x. 
Formulae (1) and (2) are taken from Dr. Hart* who has treated 
the cases of squares and cubes at some length. Formula (3) 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: 
2 ^ O' ® +1) (2-'« +1); 
Sx. 5 = tV (-^ +1)' (2^-’+ 2.r -1). 
I. »(= 2: 
Ex. 1. Using formula (2) assume a; = 5, ^ = 6, and ^ = 1. 
Then d = 42, which by trial is found equal to 
12 42 _|_ 52 ^ Yvhence 
2^+3‘'^+6'^ = U. 
*See The Mathematical Magazine. Edited and published by Artemas 
Martin. 4°. Erie, Pa. 1882-1884. Yol. 1, No. 1 [January, 1882], pp. 8-9, 
and No. 11 [July, 1884], pp. 173-176. 
