258 The Rev. Brice Bronwin on M. Jacobi’s Theory 
mitted to remark, that several interesting inquiries of a kin- 
dred nature are suggested by them: the prosecution of these 
I propose to publish in « distinct form, as a supplement to 
the volume already referred to. 
J. R. Youne. 
XLIII. On M. Jacobi’s Theory of Elliptic Functions. 
By the Rev. Brick Bronwin*. 
[8 page 36 of his Fundamenta Nova, &c., M. Jacobi, making 
PCIe, cag: 
wo = MK + mK V1 says that m and m! may be any 
2 is 
integer numbers, positive or negative, provided they have no 
common factor, which also measures 7. What I intend in this 
paper is, to prove that m must be an odd and m! an even 
number, and that no other form is admissibie. If7 and 7! be 
integers, positive or negative, the value of w, as above defined, 
includes the four following forms :— 
ie (2r +1) K+2" Kk’ —1 
ae n 
ow 2K ter Ky -1 
n 
)Sinamnw =+1,cosamna = 0. 
,snamnw = 0, cosamna=+1. 
= QrK+ 2r+I)K f=1 sinamnw = + wa/—l,cosamna =-+ o, 
n 
a= Br Dir iirS nT sinamne= cosa m na = +4 /1. 
The values of sinamnw, cosamnw are annexed on ac- 
count of their importance in what follows. ‘They are calcu- 
lated by the formule at pages 32 and 34, The references 
here made are all to the Fundamenta Nova, and the notation 
adopted there is empleyed here, unless express mention be 
made to the contrary. But to abridge I shall write sau for 
sina mu, cau for cosamu, &e. 
At page 38 we have for the general transformation, 
saag= Csausa(u+ 4w)sa(u+ 8w)...sa (u + 4:(2—1) o),. (1.) 
or 
ay a2 me mw ) (1 a? ‘ ) 
u M J ieade ~ 2a8a) 7 sa2(n—l)os ., (2.) 
840 ~ GA — Patsta4a) (1 — sa? s?a8.0)...(1— 4 a? 8? a2 (n—1) 0) 
Ass : 
Here — is put for the constant denominator of the second 
C 
member of (1.) The middle factor of that member is 
* Communicated by the Author. 
