392 Proceedings of the Boyal Society 
where, to prevent degeneration of the cycle, we must have 
ie., {q 4- 2)m - + 1)^'> 2^' 
and 
M>g. 
Writing N for m - 1, this special result becomes 
sj { 5(«+2)N }"-N= I ife+2)N-l |"+- 
wliicb is the accurate form of M, de Jonquieres’ second theorem. 
M. de Jonquieres’ oversight consists in omitting to notice that the 
g in his statement mu-st be less than <2 + 1, 
6. Theorem III. is avowedly a combination of theorems I. and 
II. ; it is carefully stated, however, in such a way as to seem to 
support M. de Jonquieres’ theory that the number of terms in the 
cycle of the continued fraction for is dependent upon the 
ratio of : d, 
7. Theorem IV, there is no accounting for ; it is but a case of 
theorem II., viz., where & = 2a + 4 and e == 4. hleither is there any 
accounting for the remark following it — ‘‘Le nombre 12, qui 
devrait figurer en tete de la serie, fait seul exception, parce qu’il 
rentre dans le groupe defini par le theoreme I, a cause de 
12 ^3^ + 3”; for 12 is the case where ?2 = 0, and there could be 
no continued fraction with the cycle 1, 0, 1, 2a. If it be an excep- 
tion, then a whole class of exceptions to theorem II. has been 
overlooked, viz,, where q~2, 
8. When Z ~ 5, the general theorem gives 
mJ { + ‘2m +i>^ + 1)M + 4- g)(g2 + 1) j. ^ 4.^ + g)]V[ + (^2 + 1 )2 
