ME. J. W. L. GLAISHEE ON THE THEOET OF ELLIPTIC FUNCTIONS. 491 
» 1 i , 7 T U , Ti ll t , 7TX’ VX' 
Also let ^=2K an( * z ~oW’ so ^=— =— . 
§ 4. The process of transformation into the form 
<p%db<p(x— |W')± , P(^+jM')+ & c - 
may be conveniently exhibited on (2) ; we have 
2Kx 
cos am ~^~ = cos am u = sec am (ui, k'), which, from (6), 
'2/cK')e z - 
4 r ez + e 
TT~r 
2 1 
4r s e? z + e~ 
&C. 
: /JIC j ez+e- 2 ~( eZ ~^ e ~ z )( r — r 2 -{- r 3 — &c .) + {& 3z + e 32 )(r 3 — Z+r 9 — &c.) — &c. j 
7T l 
re? 
/cK'|e z + e z 1 + r 2 e 2z 1 + r 2 e _2z I + r 4 e 2z- ^ 1 + r 4 e -2, 
1 1 
-&c. 
’&K')e z -(-e -z re z + r -1 e -z 
l 1 
1 1 _ , 1 _ &c l 
+ re~ z ' rV + r~ 2 e _z ' r 2 e 2z + r 2 e 2z j 
'AK' 
■ + r *r r* 1 +r 0 ^ rir + 1 -J-j* C +1 ) rw 2 + r 0 0 rn ,+2 +/‘ C*- +2 ) j 
:+- 
77-S— &C.1. 
The process requires that re z should be <1, that is, that u should be <2K; but as 
both sides of the equation are such that they change sign without being altered in value 
when u + 2K is written for u, we see that the result obtained is true for all values of u. 
Thus we have 
cos am 2K.r=- 
' kK!\r* +r~ x r* -1 +r - (* -1) 
for all values of x. 
If in (10) we take x=0, we have 
7r 2 r 
4=g^+ ^-»+ r -«-« + &c -} • • (io) 
+ r~^ _ r" 
■ + r 2 
■ — &C. 
or, writing K and K for K' and k, and therefore q for r, 
2h'K 
ic 
4g , 4g 2 
1+g 2 M + r/ 
which is at once seen to follow from (7), and is given by Jacobi, ‘ Fundamenta Nova,’ 
p. 103. 
It is, of course, easy to deduce (9) directly from the infinite product 
1 — cos am u s 
1 + cos am u 
=tan 77 II 
(1 — 2 q 2n cos x+q * n ) (1 + 2q 2n ~ l cos x+q in ~ 2 ) 
(1 + 2g ,2,t cos x+q in ) (1 — 2 q 2n ~ l cos x + q in ~‘ 2 ) ’ 
for consider 
1 — 2q 2n cos x 4- q in . . , (1 — q 2n e ix ) (1 — q 2n e~") 
1 + 2 q 2n cos x + q* n ’ W 1C (1 + q 2n e ix ) (1 + g ,2 “e _ix ) ' 
