f 
490 
ME. J. W. L. GrLAISHEB ON THE THEOEY OE ELLIPTIC FUNCTIONS. 
(in which, after the first term, the series proceed by pairs of terms, so that for every 
term — - — there is a term — r — 
x — me x + m r 
Thus in general (although the sine and cosine are, as just mentioned, exceptions) 
we shall have, by equating the different forms of \|/#, identities such as ex. gr. (if 
is even) 
Q'KOC 4l7TX 
<px-\-<p(x—yj)-\-<p(x+yj)-\- &c.=Ai+A! cos — +A 2 cos— — j- &c. 
Also, it will be seen in § 10 that in certain cases even when ipx is not periodic it may 
be exhibited in the form [*)-\-<p(x +(*)-{- &c., and we shall obtain identities 
in which the two sides of the equation are non-periodic. 
§ 3. Before applying these principles to the elliptic functions, it is convenient to 
write down at once the following eight formulae, which are to be found in the ‘ Funda- 
menta Nova ’ (pp. 101, 102, &c.), and which are all placed together in Durege’s ‘ Theorie 
der elliptischen Functionen’ (Leipzig, 1861), pp. 226, 227 : — 
2 tt f qi .7 TU qi . 37 ru , 0 ) 
sin am sm p^p sm ^ + &c. |, 
2tt ( q Ttu qi 3mi 0 ) 
cosam«=j E | n ^co 8 2K+ TT? cos2K+ &c.j. 
cosec am u 
to 4 q .to 4 q 
= 2Ki COSeC 2K + T=5 Sln 2K 
4 q 6 . 3to p f 
r^3 sm 2K+ &C ‘j> 
7 r f to 4 q to 4 o 3 3 to „ 
secam M = 2 FS |sec 5s - rj ^co S2 -g+ T ^cos 2 K - &c. 
1 7T 4*7 TO 4o 2 2 to d ) 
A — 77Trf?{ 1 — r~ ; — oCOS-^H- , , 4 COS -xf — &C. >, . 
Aamw 2&'K) 1 + <p K 1 1 + g 4 K j’ 
7T ( ,7 TU 4 fl 2 . TO 4<7 4 . 2TO n ) 
cot am w=2ld cot 2K — 1+F Sm K _ 1+7* sm K ~ &c< 
( 1 ) 
( 2 ) 
. 7T f 4 q TO 4o 2 2 to n ) 
A am m=2k| 1 + cos k+TT? C0S X + &c j’ ( 3 ) 
, 7T f TO 4^ 2 . TO 4g 4 . 2 to d ) ... 
tan am ^^tan ^-pq^sm K+p^srn &c.|, (4) 
(5) 
( 6 ) 
( 7 ) 
( 8 ) 
wherein, of course, K . 
_7tK 
In what follows, let r—e~ K ' , and take 
ttK' ttK 
P=~K » "= X’ 
so that 
M , r=e ", and g J v=n <1 . 
