492 ME. J. W. L. GLAISHEE ON THE THEOET OE ELLIPTIC FUNCTIONS. 
Taking the logarithm and differentiating, we obtain, after a little reduction. 
wif 1 
K| ‘ q~* n e™—q* 
o—ix I n— 2 
Similarly, from the uneven factor we get 
.«/. 1 . 1 
g 2 * »— 1 g“ 5 q— (2»— l)g— i»l q—(2n—l) e ix g2n— lg- 
thus 
k! m y( 1 
>s am (K— u) 2K C0SeC ^~’ _ K q-^e -1 
2n p ix n 2n p —ix ry 2n ~ l P ioc n -(2n-l) p -i 
q—**e — q*n— L e ™ — q 
q-(2n-\) e ix_q2n-\ er ixy 
Replace u by K — u, that is to say x by \k — x, and remembering that e^—i, 
e -¥*— we fi n d 
secam«=~| S ec*+22( ?iei ~=^+. . . ■ ■ •)}• 
(9) for sec am (ui, k'), that is, for cos am u. 
If the other formulae in the group (1) to (8) be transformed in the same way, viz. by 
use of the identical equations 
sin am u——i tan (m, k’), 
A am u= cosec am (ui-\- K', kf), 
we obtain the following seven formulae 
-(z-l) ^+i_ r -(r+i). 
0 7T (r* — r~ x r x ~ 1 — r -(x-i) ^+i 
y X 1 _|_ /* ( X 1) ryX+lj^y* (£+1) 
yX — 2 rp— (X— 2) ^.Z + 2 y— (Z + 2) 
— &c. 
“h pX — 2 ( x ~ 2 ) /-Z+2 J.— (Z+2) 
A am 2 K# = i } -f ^ + 1 + * - (*+ 1 > + &c, 
tan am 2K^=^,| rZ _i_ I r _ ( ,_ i) + ^ +i 4-(x + i ) + ?J - f 4--^f )+ &c 7’ 
1 7 T jr r + r- x i*- 4 -f r - ^- 1 ) ,*+i + r -(*+i) | 
sin am 2 Ka? 2K , |r c — r - ® r 1-1 — r c+1 — r - ( :c+1) "^’ C 'j’ 
1 7r f / ‘L ■- I ■ 1 ) 
cos am 2Ka' k!}sJ\r x ~'^ — r _ o-3) r x+ ^ — r x_ |- — 7 ,_( - r_ S ) '^ - C 'J’ 
A am 2K« ^K , "jr r_ ^ + r~ (z_ ^ ) 4"^+! + ,--^+*)+ + r -o-§) + & c - 
• (11) 
• (12) 
• (13) 
• (11) 
■ (15) 
• (16) 
cot am 2Kr= — ^| y Ji~^4- ^:- 1 _^- (a: -i ) + y . J+ i_ 1 r - (a ;+ r ) + &c.j. . . . . (17) 
