414 SIE F. POLLOCK ON FEEMAT’S THEOEEM OF THE POLYGONAL NUMBEES. 
roots having the same sum but the ditferences reversed, and the sum of their squares 
2 7 4 
4+254-36 = 65: therefore 65 = 2, 5, 6. Again, —5, 2, 6 have the ditferences 7, 4; 
4 7 
their sum =3; but —4, 0, 7 have the differences reversed and the same sum; therefore 
2 
-5, 2, 6 = -4, 0, 7=65, and 101 = 0, 4, 6, 7. 
The proof of this theorem will appear from putting the general case algebraically, 
Avhich also will show the method of obtaining the new roots required. Let the dif- 
ferences of the roots be represented by a, a-\-%n (which include every case): then 
[p), “■^^”(^+2« + 377) will represent any 3 roots having the requu’ed differ- 
ences: the sum of these roots is 3j) + 3a+377 [_a multiple q/* 3] : reverse the ditferences 
and take^ as the first root, and they will be —p, “■^®”(^+«+3?i), “p+2a+3?i: the sum 
will be 3j9 + 3«+6?i [also a multiple of 3]: therefore the difference will be a multiple 
of 3, and the sums may be made equal (one to the other) by adding or subtracting from 
each root the difference divided by 3 : here the difference is and the new roots will 
be 77, “■^^"/>+< 2 + 2 w, ''^+ 2«+2w : and if each of these sets of roots be squared and 
added together, the sum of each will be 3p^+5«^+9?7^+6ap + 677^ + 12«?7. 
A similar theorem belongs to 4 roots whose differences differ by 4 : thus 1, 2, 7, 16, 
as roots, have the differences 1, 5, 9: their sum is 26: — 3, 6, 11, 12 have the differ- 
ences reversed, 9, 5, 1 : and their sum also equals 26 : and 
2 2 2 2 
1 5 9 9 1 5 1 5 13 13 5 1 
1, 2, 7, 16=-3, 6, 11, 12 = 310: so -6, -5, 0, 13 = -12, 1, 6, 7 = 230, 
the sum of the roots in each case being equal, and the differences reversed. 
A similar theorem also belongs to 5 roots whose differences differ by 5, and no doubt 
to n roots whose differences differ by n. 
There are many arithmetic series of the 2nd order which, beginning with 1 as a first 
term, will have all their terms divisible into not exceeding four squares : there are 3 
such series to which I wish to call attention. If 1 be increased by 2, 4, 6, 8, &c., the 
2 
(77+l)th term of the series is always 77^+77+1, or 477^+277+1, that is, 77, 77, 77 , 77 +I. 
2 
If I be increased by 2, 6, 10, 14, &c., the (77+l)th term will always be 0,1,72,72. 
2 
If 1 be increased by 4, 8, 12,16, &c., the (77+l)th term willbe 0, 0, 77 , 77 +I. The first 
of these series contains the numbers which, being divided into 4 squares, give the sum 
of the roots a maximum : the others give the differences between 2 roots a maximum, 
the one the even differences, the other the odd differences. 
But if any odd number (instead of I) be made the first term of the series, some 
remarkable consequences ensue. If any odd number 477+1 be increased by 2, 4, 6, 8, 
&c., the term whose index of place is the lesser moiety of the odd number will be com- 
posed of 4 squares, whose roots will be the result of again dividing the moieties of the 
odd number: thus 477+1 = 277+1 + 277=77+1+77+77+77 : if the number be 477—1, 
2 
the (277 — l)th term will be (277 — 1), 77 , 77 , 77 : if it be 477+1, the 277th term wiU be 
