ALLUDED TO BY EEEMAT. 
629 
p-\-l and^)+ft— 1, the sum of the roots will then be diminished by 2ft— 2. The alge- 
braic sum of the roots is not altered by this increase of one and decrease of another of 
them by the same number. 
For, if 
or 2ft-|-l, 
then 
±«±(^ft)+(c+ft);±;^= 1, or 2ft-|-l. 
All these changes reciprocate ; thus, if a number has two roots equal, the number which 
is greater by 2 will have two roots differing by 2 ; and vice versa , if a number has two 
roots differing by 2, the number which is less by 2 will have two roots equal ; thus, if 
13 has two roots differing by 2 as 0, 2, then 11, which is less by 2, will have two roots 
equal, viz. 1,1. 
If an odd number has roots differing by 0, 1, 2, 3, 4, 5, &c., the four roots of the odd 
numbers, next in succession, one after another, will be discovered till the differences are 
exhausted ; and if every number has this property, the succession will continue through 
the whole series of odd numbers, and every odd number will be composed of four squares, 
and therefore every number will be so likewise ; for every even number is ultimately the 
double of an odd number, and the double of four squares is itself composed of four 
squares. 
There is another mode of altering the roots, substantially the same, but occasionally 
applicable when the other is not. It is mentioned as Theorem A in page 313 of the 
Philosophical Transactions for 1854 ; but I was not then aware of its full effect, and did 
not pursue the subject. 
If the difference between the sum of two of the four roots, and the sum of the other 
two be ft, and each of the larger be decreased by ft, and each of the smaller e bincreased 
by n, the increase in the sum of the squares will be 2 n, if ft=l the increase is 2. 
This is a proper place to mention a property of the series mentioned above, 1, 3, 7, 
13, &c. In a former paper (see Philosophical Transactions, 1854, page 315) I called 
this a gradation series , but I was not then familiar with its properties, and did not pursue 
the subject as I propose now to do. Every term in the series has roots of the form 
ft, n, n, %±1. If any term, as 13 (which is composed of the squares of the four roots 
1, 2, 2, 2), be increased by the methods above mentioned, taking care to keep the sum 
of the roots always equal to the number 7 (the sum of the roots of 13), it will be found 
that, from 13 to 21 inclusive, every odd number will be composed of four squares, the 
sum of whose roots is 7. In this case the process cannot be carried beyond 21 ; for 23 
being of the form 8ft -f- 7, cannot be composed of less than four squares: 5 is too large 
to be one of the roots, and four is unavoidable, since 23 — 16 leaves a remainder of 7, 
which cannot be expressed by less than four squares. The only form, therefore, of roots 
for 23 is 1, 2, 3, 3, and their sum is 9. But the sum of the roots from the number 21 
to 31 will be 9 ; and from 21 to 31 the odd numbers can be made of four roots whose 
sum shall in every case be 9 ; and this goes two steps beyond 31 ; and as the number 
4 R 2 
