of Edinburgh, Session 1883-84. 391 
2k + 2k+-' 
* * * 
5. When Z = 4 the general theorem gives us 
V{ Wm + 2g'i)M -- + 1)^2 
1 1 1 
= { } + 
1 
2'i+ ^2+ } + 
* * 
(IF) 
where for shortness there is pat 
{ } for {|(?fe + 2g'i)^-2fe2'2 + f)2'2}- 
Here again a little consideration, based on the knowledge that 
the number under the root-sign must be integral, serves to show 
that there are two distinct cases, viz., 
(1) when ^2 is even, 
(2) when is odd, odd, and m even. 
Putting therefore in the first case ^ 2 = and in the second case 
^2 = 2s - 1, q-^ = 2r-\, M = 2n, we have as our pair of identities 
J { (%" + ?)M - (2Z^-h 1 )^ } " + + 1 )M - 4^3 
111 
+ w 
q + 2k + q +2| j + 
2/1' 
M>-- 
(II.)/ 
I (8r2s-8r5- 4r2 + 8r-l2s-3)N - (2r5-r-s + l)(2s - 1) j- -}-2(2'F5-r-54-l)N - (2s- 1)- 
f ) _i 1 w 1 
= \ f +2r-l4-2s-l + 2r-l-F2{ } + -:^ 
These give every integer whose square root has a four-termed cycle, 
and make known the cycle as well 
Returning now to (II') and putting g'i = 1, we have 
\/{i(2 + 2 )m - 1(2 + l)s}2 + (2+ 1)M - 2^ 
or 
^{2(2 + 2)“- 4(2+ 1)2+ 1}^-( m -2+1) 
V{i(2 + 2)(M-2 + l)F-(M-2 + l) 
( 1 2 1 1 1 1 
= I J(2 + 2)(m - 2 + l)-l I +Y+^+i+2{ }+... 
or 
