110 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Ex. 2. Using formula (3) assume «= 100 and 6 = 212; 
whence r = 30370194, which by*trial is found 
equal to 
1*-+ 2+ 3*-+ 44+ 8+ 10*+ 14*+ 24*-+ 4244721; 
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*. 
Be is) aa / 
Ex. 1. Using formula (2) assume = 10, p=11, and q=1; 
whence d = 133 044, which by trial is found to equal 
15+ 2°+ 3°+ 8+ 10°, and therefore 
445+ 6+7+ 99+ 11? = 12°. 
Ex. 2. Using formula (1) assume p = 29 and g=1; whence 
a = 3788851, which by trial is found equal to 
5°>+10°+ 11°+ 16°-+ 19°; hence 
5+ 10°-+ 11° + 16°+ 19° + 29° = 30°. 
Ex. 3. Using formula (2) assume «=18, p=31, q=1; 
whence d= 1 731 920, which by trial is found to 
equal 1°+2°-+ 4°-++ 5°+ 9°+12°+17°, and therefore 
3+6+ 77+ 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 Sz,n for n=4 and n= 5 were tabulated. 
This paper was discussed briefly by the Chairman and by Messrs. 
DoouirrLe, Hitt, BAKER, and KUMMELL. 
