MATHEMATICAL SECTION. 
109 
Ex. 2. Using formula (1) assume p = 8 and q = 3; whence 
a = 57, which is by trial found equal to 2^ + 
4^+ 6 ^ and 
V-i-2^-i-4^+6^-h8^=n\ 
Ex. 3. Using formula (3) assume x ~ 50 and h — 206; then 
489, which by trial is found equal to 
12 4 . 22 + 22 ^ or 52 + 8 ‘^ 4 - 20 ^ etc.; 
whence we have two sets of 47 numbers, the sum of 
whose squares is a square. 
Ex. 4. Using formula (3) assume x= 100 and 5 = 1750; 
then r — 2850, which by trial is found to equal 
4^ -h 5^^ + 53^; whence 
r+2‘^-f 3^-t-6^+r4-... 
+ 52^+54^4-_4-100^ = 1750^ 
II. 7i = 3: 
Ex. 1. Using formula (2) assume x — 5, p = 0 , q==l, whence 
d = 134, which by trial is found equal to 1^-f 2^4- 
whence 
33 434 . 53 ^ 03 ^ 
Ex. 2 . Using formula (1) assume jo= 8,5 = 1 , whence a = 217, 
which by trial is found to equal l'‘^ 4 - 6 ^; hence 
U-f 3^4-4=^+ 8 =^ = 91 
Ex. 3. Using formula (3) assume x = 100 and b = 294; 
whence r = 90 316, which by trial is found to equal 
1 ^ 4 - 6 "+ 11 ' 4 - 21 ^ 4 - 43^ whence 
2 '+ . . 5^4-7^4- . . 10 ^+ 12 ^+ . . 
20' + 22'4- . . 42' -f- 44' . . + 100' = 294'. 
Ex. 4. Using formula (3) assume .r=1000 and 5 = 6303; 
whence r=5 869873, which by trial is found to 
equal 1' + 2' 4-10' 4-16' + 32' 4-180'; hence, 
3'4- . . 9'4-lU4- . . 15'4-17'4- . . 
31'4-33'4- . . 179' 4 -181'4- . . 1000'=6303'. 
III. n=4: 
Ex. 1. Using formula (2) assume rr = 10, ^ = 14, and q=l’, 
whence d= 13 124, which by trial is found equal to 
1^ 4_ 2 ^ 4 - 3* 4- 5^ 4- 7* 4- 10^ whence 
4^4-6*4-8^4-9*4-14^ = 15\ 
