Ex. 4. 
II. n=3: 
Ez. 1. 
Ex. 2. 
Ex. 3. 
Ill. n= 4: 
Ex. 1. 
MATHEMATICAL SECTION. 109 
. Using formula (1) assume p = 8 and gq = 38; whence 
a = 57, which is by trial found equal to 17+ 2?-+ 
47+. 6? and 
1+ 2?+ 47+ 67+ 8? =117, 
. Using formula (3) assume z = 50 and 6 = 206; then 
7 = 489, which by trial is found equal to 
1? + 2? + 22 or 5?+ 8?-+ 20, etc.; 
whence we have two sets of 47 numbers, the sum of 
whose squares is a square, 
Using formula (3) assume z= 100 and 6=1750; 
then r = 2850, which by trial is found to equal 
4? 5?-++ 53?; whence 
V+ 2+ 34+ 6477+... 
+ 52’ 547+... . +100°=1750? 
Using formula (2) assume «= 5, p= 5, g=1, whence 
d = 134, which by trial is found equal to 1°+ 2°-+ 5%, 
whence 
3+ 44 53 = 6% 
Using formula (1) assume p =8, g =1, whence a =217, 
which by trial is found to equal 1°+ 6°; hence 
13+ 33+ 48+ 58+ 8 = 9, 
Using formula (3) assume «= 100 and b= 294; 
whence 7 = 90 316, which by trial is found to equal 
1°-+ 6+ 11°-+ 21°+ 43°, whence 
J+ ..5+774+ ..10°4+12%4+ .. 
20° + 228+ .. 42?+ 444. . + 100° = 294°. 
. Using formula (3) assume «=1000 and 6= 6303; 
whence r= 5 869 873, which by trial is. found to 
equal 1° + 2° + 10°-+ 16° + 32°+ 180°; hence, 
a+ ..9+11°5+ ..15+174.. 
313+ 33°-+ .. 1797+ 181° + . . 1000°= 6303%. 
Using formula (2) assume e=10, p=14, and q=1; 
whence d = 13 124, which by trial is found equal to 
“144+ 2+ 3+ 5'+ 7+ 10*, whence 
4* + 6+ 8*-++ 9+ 14*= 15+, 
