of Edinburgh, Session 1883 - 84 . 
395 
(1) a odd, c even. 
(2) h even, c odd. 
(3) a even, c even, m even. 
(4) a even, b odd, c odd, m even. 
As a particular case of the first of these particular cases, let us 
take a = 1, c = 2, m - 25^ -j. l. Then 
This is M. de Jonqui^res’ Theorem VI., and was given by me as a 
chance example in 1874 (see p. 31 of The Expression of a Quadratic 
Surd, ^'c.). 
11. Having enunciated these six theorems, M. de Jonquim^es 
adds : — “ Les theoremes I, II et III montrent, comme je I’avais 
aiinonce, que la longueur et la composition de la p^riode dependent 
principalement de la valeur du rapport 2a -~d quand cette valeur est 
entiere, et tons mettent en evidence ce fait que, dans une meme 
famille de nombres, ceux des termes de la periode qui changent 
d’un nombre k I’autre sont de la seule variable.” This mode of 
reasoning is somewhat perplexing. Being concerned with functions 
of any number of variables, M. de Jonquieres takes the functions in 
the ultimate or penultimate stage of specialisation, and tries to draw 
general conclusions from the results thus obtained. Such a course 
could not but be futile. Knowing, as we do, that, g 2 » • • • » S'a? 9i 
being the terms of the symmetric portion of the cycle, the expres- 
sion for M. de Jonquieres’ 2a is 
K(gi,g2» • • • » ^ ?3^^2)K(g2» • • • ^^2) > 
and for his d, 
we have instantaneously forced on us the conclusion that for the 
solution of our problem something besides the ratio of 2a to d must 
be taken into account. Hot only, however, does M. de Jonquieres 
merely direct his attention to very special cases, but these special 
cases are specially selected ; special cases that tell a different story 
are ignored. What simple peculiarity, we may ask, of the ratio 
2a ; d exists fnr the infinite number of other special cases of our 
1111 1 
l + ^+ 2-}-6-l-l-{- 4(6 + 1) -j- . .. 
K(?1, . . . ,?2)“ - ( - • • • >?2)^ . 
