316 SIR F. POLLOCK ON CERTAIN PROPERTIES OF SQUARE NUMBERS, ETC. 
any positive integer r' (not greater than r) be assumed, it will always be possible to 
satisfy the pair of equations 
w‘ 2 =x i -\-y 2 -\- z 2 =2n -\- 1 
w-\-x-\-y-\-z = 2r ' -\- 1 
by integral values (positive, negative, or nil) of w, x, y , z. 
I now propose to show in what manner the table may be used, so as to divide into 
square numbers (not exceeding 4) any odd number from 1 up to 9503, and any odd 
number whatever of the form 4w 2 +2/z+2p + l, where p is not greater than 95 [95 x 
2+1 = 191]. 
If 2 j 0 +l=a 2 + 6 2 +c 2 +cP, and if also « + &+c+d= 1 ( a , b, c, d being integral num- 
bers, positive, negative or nil), in other words, if the odd number 2p+l be such that 
the algebraic sum of the roots of the square numbers (not exceeding 4) which compose 
it may be equal to 1, then it will follow that m 2 + m+l+2/; may be resolved into 
square numbers (not exceeding 4) the sum of whose roots will equal 2m+l, for m 2 -\-m 
+ 1 is of the form 4« 2 +2?z+l ; let it equal it, and ra 2 +m+l+2p=4w 2 +2w, (a+Z> 
+ c+c?) + a 2 + 6 2 +c 2 +e? 2 = (n+a) 2 + (rc+&) 2 + (w+c) 2 + (n+d) 2 (manifestly 4 square 
numbers), and the sum of the roots=4w+(a+6 + c+c0 = 4w+l=2m+l. 
Let the function ra 2 +ra+l be designated by the notation fm. If every odd num- 
ber from 1 up to 2m+l can be resolved into (not exceeding) 4 square numbers, the 
algebraic sum of whose roots may equal 1, then every odd number from fm to fm 
+ 2 m inclusive may be resolved into 4 square numbers, the sum of whose roots may 
equal 2m+l ; but the next odd number to fm-\-2m is /(m+1), and since /'(ra+1) is 
resolvable into 4 square numbers, the sum of whose roots=2?w+3 ; if every odd num- 
ber from 1 up to 2ra+l can be resolved into not exceeding 4 square numbers the 
algebraic sum of whose roots=l, then every odd number from 1 up to f{m-\- l) + 2m 
is resolvable into 4 square numbers, and t /'(ra+l)+2m=ra 2 +5m+3. 
In the Table the highest odd number 191=2x95 + 1, therefore m=9 5 ; and every 
odd number from 1 up to 95 2 + 5 X 95+3 = 9503 may be resolved into not exceeding 
4 square numbers, by means of the Table, also every odd number of the form 4n 2 +2» 
+ 2/>+l, whatever be the value of n, provided^ be not greater than 95 ; for example, 
let it be required to resolve 9301 into 4 square numbers, the next less number of the 
form m 2 +m+ 1 is 9121=95 2 +95 + l=4-48 2 — 2*48+1, 9301=9120+181. 
181 by the Table is resolvable into 
l 2 + 4 2 +8 2 + 1 0 2 
and -1+4 + 8-10=1 9301 = (48+l) 2 +(43-4) 2 +(48-8) 2 +(48+10) 2 
= 49 2 +44 2 +40 2 +58 2 ; 
so 4w 2 +w+181 is always resolvable into 4 square numbers, whatever be the value of 
7i, and the roots of the square numbers will be (rc+1), (n+4), (w+8), (w+10). 
