derived from Elliptic-Function Formula . 541 



and the denominator 



\/q 3 q 3 \/q 3 5</q b _ 7qWq 7 , « 

 l+q* + ~l+tf l + q"> 1 + ^ 4 +6CC -' 



the numerators of the fractions being alternately of the forms q n 

 and q n \/q n . 



There is another expression for tan yjr which is also worth 

 noting. Putting qi for q in Euler's identity 



l-q.l-q\l-q 3 .l-q 4 ...=l-q-q' 2 + q 5 + q 7 -q l2 -8zc., 

 we find 



arc tan q— arc tan q 3 -f arc tan q 5 — &c. 



q — q 5 + q 7 — q l5 — 8zc. 

 l+g a — 5 12 — ^«— g 96 — &c* 



the exponents being the uneven pentagonal numbers in the nu- 

 merator, and the even pentagonal numbers in the denominator. 

 In the numerator the sign of ? N (N = i {Sn 2 ±n)) is (-)"+**-», 

 and in the denominator (__J n+ * N e 

 II. In the formula 



_ / l-g.l-g 3 .l-g 5 ..A 4 



put V q for <7, and 



1-ft _( l-\ / q *\-s/q 3 . 1- Vg 5 . . A 4 



i + *~Vi+^.i + V? 3 .i+^ 5 .../ ; 



change the sign of q } and 



= arc tan 



tt—ik ( \ — is/q . l + zVg 3 . 1— iVy 5 . . A 4 

 k' + ik \1 + iVtf . l-i\/g 3 . l + »Vff 5 ..'./' 



whence 



k 

 arc tan y =4(arc tan \/q — arc tan Vq 3 + arc tan \/q 5 — &c), 



A; 

 which, since arc tan -n — 0, the angle of the modulus, is Jacobi's 



formula, "quae inter formulas elegantissimas censeri debet" 



k 

 (Fundamenta Nova, p. 108). Replacing q by # 2 , arc tan p- be- 

 comes 



arC n 2VA' = 2 ~ 2 arc tan ^ > 



