( 429 ) 
put equal to p(2r+ 1) gives n= EN =f 79 Vp? + Bag p — Sho, 
b 
so that p must be greater than mae 
0 
The other factor gives 
: p 1 ie NRA LN 
hen UP Say Bi; Bbq, 
so that the form under the root sign proves to be in both cases 
the same. 
VIII. Use of the series 1 4 3 + 5 + etc. 
A square a? is equal to 1 +3 4-5 H...(2a — 3) + (2a — 1), 
To denote a non-square, e.g. 953 (= 30° + 53 = 312 —8) in this 
manner, let us write that series in the form 
jE Bes ESS BBO (AREN Ie 64 Ots | 
According to the method of § I we write: 
953 = 312 — 8 = (312 + 63) — (8 + 63) = 32° — 71 
— (82? 465) — (71 + 65), etc., 
until the subtrahend has become a: square. 
The series gives a clear representation of that operation. 
But if 8 + 63-4 65 +... is a square, then it is possible to build 
it up out of 14+ 3-+4 5 + ete. and this gives the following operation : 
subtracted remainder 
8 
1 7 
: a 68 The method is laborious, whilst there 
5 55 Is a great chance for errors. However, 
9 46 it gives a clear insight into the composi- 
11 35 tion of the numbers and the writer is or 
a 2 465 opinion that probably from this method later 
17 to 21 investigations will start. 
or 3 X 19 | re 
3 X 25 7 + 69 
2 X 30 16 + 71 
ete, 
