260 ‘The Rev. Brice Bronwin on M. Jacobi’s Theory 
between these values. Consequently, while x increases of 2 w, 
a“. : Jon F 
M increases of 2H, neither more nor less, if —. = H when its 
M 
; ; : u 
amplitude is > Whilst, therefore, « from 0 becomes a, M 
from 0 becomes H. Let them have these values in (3.), and 
we obtain saH=1= + Csawsa3aw...sa(2n —1)o 
Therefore 
oF + SA2HWSA3wWSA5 Wer Sa(2u~—1)w. . (5) 
> and M. Jacobi’s de- 
nominator cannot be true, except in those cases in which it is 
reducible to it. i 
One factor of (5.) issanw. The second and third forms (A.), 
u 
therefore render C faulty, and also the values of s a Ww of M, 
and of the modulus of M faulty ; for C enters into these values. 
If M. Jacobi’s formulz do not fail for these values of , jt is 
This then is the general form of 
because they do not hold true for them. His value of on is 
C 
{sa(K — 40)sa(K — 8a)......sa(K — 2(m — 1) )}2 
a eee A 
~ LA.a4mA.a8o......A.a% (n—1) 0 
i J 82.06.24: w..inoes cata — Ne Me 
~ LA.a2uh.a4a......A.a(n—1)HJ ’ 
whatever be the form of w But for the first of the forms (A.) 
only can we have 
SAWSA3 W.e.SA(2N—1) w= {Sawsa3w...Sa(n —2)w}* 
oa Se EE em 
iit A.d2WA.G4W oer A.a(n—1)w z 
For the other forms of w this reduction cannot be effected. 
M. Jacobi’s formulze therefore are only true for the first form 
(A.) In page 39 we have 
u  cauca(u+4w)ca(u+8w)...ca(u + 4(n—1)w) 
ag {ca4wca8w...ca2(n—1)m }*? (6.) 
This reduces as before to 
u_ , cauca(ut 2w)ca(u+4w)...ca(u+2(n—1)w) 
cays {ca2wea4tw...ca(n—1)w}* ‘oe 
