110 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Ex. 2. Using formula (3) assume x~ 100 and h = 212; 
whence r= 30 370194, which by trial is found 
equal to' 
1^ + 2^ + 3* + 4^ -h 8* + 10^ + 14* + 24* + 42*+ 72*; 
hence if all the natural numbers from 1 to 100 , ex¬ 
cept the ten above given, be raised to‘the fourth 
power and added, their sum equals 212 *. 
IV. ^^=5: 
Ex. 1. Using formula (2) assume x = 10, = 11, and 5 = 1; 
whence d = 133 044, which by trial is found to equal 
1 ^ + 2 ^ + 3^ + 8 ^ + 10 % and therefore 
4^+5" + 6 ' + 7' + 9' + lU-= 12 % 
Ex. 2. Using formula (1) assume = 29 and ^ = 1; whence 
a = 3788 851, which by trial is found equal to 
5^+10^+1U+16^ + 19^; hence 
55 lo- 115 q- 165 -}- ^95 ^ 29^ = 30% 
Ex. S. Using formula (2) assume = 18, p = 31, q—l; 
whence d~l 731 920, which by trial is found to 
equal 1^+2^+4^+5^+9^+12^+17% and therefore 
3 ' + 6 ^ + 7 ^ + 8 ' + 10 ' + 11 ' + 
13' +14' + 15' + 16' +18' + 31' = 32% 
In finding these numbers, use has been made of Barlow’s tables of 
fourth and fifth powers. To further facilitate the work the values 
of See ,n for 71 = 4 and n~ 5 were tabulated. 
This paper was discussed briefly by the Chairman and by Messrs. 
Doolittle, Hill, Baker, and Kummell. 
