of Edinhurgli, Session 1883-84. 
583 
he found y then another solution is 
( 7 s ’ > • • • > ^' 2-2 ~ ^ 2-2 ) ' • • > ^'^4 ) % • 
When 2'z .. i = 29 'j + 1, we know that g'2=lj and therefore, from 
§ 6 (yS), the equation for determining the other elements is 
fe . • • 
• ) 5 .- 2 ) = (?3 . • 
• • J (?z-2) “ (g '2 J • 
• • ? qz-z) 
i.e.y 
= (?3 . ■ 
. • . 2.-3). 
or fe,.. 
CO 
11 
1 
• • J ■ 
•>(?3-2)+((Z4?-- 
• , 2.. s)' 
hTow the substitution here of 2'3_2 » 2‘z-s >«•••? > 5's for ^'3 , ^4 , • • . 
9 ’z- 3 ?$’z -2 respectively, changes the second of these continuants into 
the third, the third into the second, and does not alter the first and 
fourth ; that is to say, the equation is symmetrical with respect to 
the two sets of quantities, which proves our theorem. 
9. In a cidminate cycle of 2z dements where q^_i = 2q^ + 1 we have 
either 
g -3 = l, and g.-g- 1 ) = 2 ( 3 - 4 , < 7 ^- 3 ), 
or 
2'3 = 2, 3 -z -2 = 2, and (^ 4 ,... s) = (?4» ••• >^^- 4 ) + fej ••• , ^ 
or 
33=2 + «,3^-2 = 1, and (1 +«,g-4, ... ,gz-d) = 2(g4,..., 
From § 8 the equation for determining ^'3 , ^4 , &c., is 
(33,..., 3^-2) = (g3»--M + + 
This evidently gives 
33(34,...,3z-2) + (2'5»---; 9'.s-2) = ?3(9'45---»2'^-3) + (2'55---5(?«-3) 
+ ('Z4.---»^?^-2)+(9'4 V , 
-j , (g o? + (^ 4? •••>$' ^- 2 ) + (g 4> 2 ) 
(?4? ■••?gz-2) ~ (g4? ••"?g«-3) 
^ 2(34,...,3^-3) + (35,..„3^ -3) -(g5...,g^-2) 
(g4?-*-?g^-2)-(g4? ••?g^-.3) 
_ ^ 2(g 4?-- ?g^-3)~(g.3?---?g.^-2- 1) ^ 
(g4?--'?g^-2)-(g4?-*-?g^-3) 
Hence (?3=1, and 2(2'4,...,<7,_3) = (g5v?(Zz-2- 1) is a partial solu- 
tion. 
