of Elliptic Functions. 259 
sa (u + 2(m —1).), and the following factors easily reduce 
to +sa(u+2u), +sa(u+ 6), &c., whatever form 
takes. Also in (2.) sa(2n — 2), sa(2n — 6), &c. reduce 
to +sa2w, +sa6,&c. Therefore (1.) and (2.) reduce to 
sax = Csausa(u+2w)sa(u+4w)..sa(u-+2(n—1)). (3.) 
Sear meh se?..) 
u M\" 7 s?a2a GQ-sa “XS sa (n— 1) . (4) 
S4y7 (1 — Pa 8a20)(1—-Paesad4a)...l—-Pesa(n—l)e 
_ The quantity C, if we give it M. Jacobi’s form, will reduce 
in like manner ; but this isof noconsequence. M. Jacobi ap- 
pears to have been aware of the reduction above effected, as 
he has partially made it at pages 41 and 51. 
It hence appears that the second form in (A.) will always 
reduce to one of the three other forms. For making a 
rK+7K'V—1 , 
= ats, we have w = 2’; and putting 2! for 
w in (3.) and (4.), these last become of the form (1.) and (2.); 
which will again reduce to the forms (3.) and (4.), #' being in 
the place of w. And if w! be divisible by 2, we may repeat the 
operation, and may continue to do so till we arrive at an 
ue DK KiV¥—1. ‘ “3 
w eee ; in which one or both the quantities p 
n 
and g are odd numbers. The same reduction might also be 
made in the values of ca M and Aa we When, therefore, 
is divisible by 2, the formule of transformation will reduce 
till w takes one of the three other forms. 
Again, the second members of (1.) and (2.) are not proper re- 
presentations of the first. For since sa2(n—1)w=+sa2u, 
sa2(n — 3) = +sa6w, &c.; these members vanish when 
u=20,u=6w, &c., and therefore ought to contain the 
2 2 
factors 1 — , &c. It must be remembered 
Fs ' a 
Sa2w  sabw 
that x =sau. For the same reason the second members of 
(3.) and (4.) would be improper representations of the first, if 
w were divisible by 2. The second form of (A.) therefore is 
inadmissible. 
For the three other forms of w, we cannot reduce sa( — 1) 
to +saw,sa(n— 3)wto +sa3u, &c.; and it is not possi- 
ble to reduce them to sines of any other amplitudes. Tor these 
values of w, then, (3.) and (4.) are in their simplest forms; their 
second members vanish when u = 0, 2 a, 4, &c., but never 
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