IN A RESISTING MEDIUM. 
593 
S*V 6 
(-»( ,+ w 
_ 1 —cn a Ik' + &cn a\ 2 
~~ 1 + cn a \^' —fa a/ ’ 
equivalent to 
1— sn (2y, ti) __ 1 —sna (\ — a sn a\ 2 
1 + sn (2y, k') ~~ 1 + sn a \1 + a sn a/ 
where a = -(^3—1); and therefore by means of the second cubic 
& 
transformation (Cayley, Elliptic Functions , p. 186) M —; and by 
reason of the modular equation, 
V** + s/k'X' = 1 : 
since, if X = P, A' — k ; 
then 2 s/icJc' = 1; 
or 2M' = ~ = sin 3o° ; 
z 
and £ = sin 15°, £'= sin 75°, as required, 
Therefore we may put 
2y —,/Sa, and 2/5 = —a, 
so that f$ + iy= — + i J\/^ a — wa , 
and P — iy = — \ a — i H/3a=w 2 a. 
Therefore 
and 
0 3 -l 
c 3 + lc 3 (cn x — cn a) (cn x — cn wa) (cn x — cn co 2 a) 
(1 
X^-A 2 
cn x ) 3 
- 1 
1 V 3 ?> 3 
__ (c 3 + P) (cn x —cn a) (cn x — cn a>a) (cn x — cn oo 2 a) 
~ 12^3 6 3 c 3 cn X )3 
0 c 3 + P (cn x — cn a) (cn x — cn wa) (cn x — cn <D 2 a) 
“ (P + k) 3 (1 — cn x) 3 
_ 2 (cn x — cn a) (cn x - cn wa) (cn x — cn w 2 a). 
(1 — cn x) 3 (1 — cn a) (1 — cn coa) (1 — cn w 2 a) 
giving the factors of X 2 — A 2 , 
