16 
Proceedings of Royal Society of Edinburgh. [sess. 
the identity becomes 
A 1 1 1 
i 
4* • • • + a z 4- 2 A + • • • 
* * 
%-,«,) (- 1)* i i 
K 0 + 2K 0 K 1 “22K 0 K 1 K 2 '23K 0 K 1 K 2 K 
The fourth term is then seen to be derivable from the third by 
merely dividing by 2K 2 , the fifth from the fourth by dividing by 
2K 3 , and so on. 
In the next place, each of the K’s after K x is easily calculable 
from the preceding K. For example, 
K 2 = K(c& 2, . . . ,<x 2 , 2A,& 4 , . . . j a zi A), 
= K(a 1 , . . a z , 2A)K(a 1 , . . a 2i A) + K{a v . . .a z )K(a 2 , . . a a A) , | 
= { 2K(a 1 , ...,a zi A) - K(a lt . . .«*_!> } K(a 1} . . .a 2 , A) 
+ K(a 1> . . .,a z )K(a 2 , . . . , a a A), 
— 2K? — K(<2 1 ,...,ct z _ 1 )K(a 1 , ...,05 z ,A) 
+ K(a 1} . . a 2 )K(a 2 , . . .,a 2 , A) 
-2Kj+(-iy > 
the last step being due to the fact that 
are really 
K(a 1? . . . 
■ , a* A) 
K(a 2 , . . 
• > a zi A x ) 
%, • • 
. ’ 
K( ai , . . 
• > a z- 1) 
KK . . 
• . «i, A) 
K(a 2 , . . 
. , A) 
K(a 1) . 
KK • 
, . , a 2 ) 
— that is to say, are successive convergents — on account of a v . . 
being the same when read backwards as when read forwards.* 
Similarly 
K 3 = 2K 2 + ( - l) 2z+1 
K 4 = 2Kl + (-lf +3 
* If this Were not the case we should have — 
— K(<x 1 , . . , a 2 -i)K(a lf . . . , dz, A) + K(a ls . . . , az)K(a 2 , • • • » ®z> A) 
= a z -i)K(a v , a Z) A) + K (a lt ...,a z , A)K(a 2 ,..., a z ) + (- l) s 
= K.($i , ... } ciz) A) ^ K($2, . . . , Ug ) — ...j d z ~ 1 ) I" + ( — 1) . 
