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Proceedings of the Royal Society 
identity (II.) besides M. de Jonquieres’ special case ? And when tbe 
ratio 2a : cZ remains constant, because of a varying as fZ, is no altera- 
tion made in tbe extent of the cycle? Surely M. de Jonquieres, 
from actual observation, knows as well as any one that if merely 
2a cZ be constant, the general ride is that the cycle varies 
in extent, and that the case where it does not vary (M. de 
Jonquieres’ theorem I.), instead of being the rule^ is the exception. 
Then, again, M. de Jonquieres forces into his service facts which 
are manifestly against him. He says theorems I., II., III. bear 
out a certain conclusion. How III., as we have seen, shows 
nothing that I. and II. do not show, and need not therefore have 
been referred to ; and Theorem II. shows something totally 
different from Theorem I. In Theorem IL the ratio considered is 
not 2a : tZ, but 2Z> : e, Z.e., 2(a + 1) : 2a - cZ -i- 1. This theorem is, 
therefore, not a support to M. de Jonquieres’ theory, but the 
opposite. 
As for M. de Jonquieres’ second fact, which, as he says, all his six 
theorems bear out, we can only meet it by asking in wdiat sense it is 
possible seriously to talk of or q^ as being functions of 
1%). 22. •• • . 22. 2i)m-( - 22. ■ • • . 22W22. • 22)- 
Had M. de JonquiOTes confined himself to refuting Lagrange’s 
statement that the extent of the cycle in the expansion of 
depends only on the value of E, he would have been on safe ground, 
for the incorrectness of the statement has long been known, and 
indeed must have been known, one would think, to Lagrange him- 
self; the only conclusion, however, beyond this, to which his 
researches entitle him, is the vague one that it depends somehow, 
as Lagrange also said, “ de la nature du nombre E.” 
12. Having, in order to follow M. de Jonquieres, considered the 
cases of the general theorem where Z = 2, 4, 5, 6, it seems desirable 
for the sake of continuity to put on record the details of the 
omitted case, viz., where I = 3. The theorem then becomes 
x/{2(2^ + l)M + i2}^ + 2M + l = { }+^+2 + 2{^+ ... 
* * 
It is readily seen, however, that in order to avoid fractions, q and 
