438 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
we have 23 0 =^j- —1; and putting as above u— 0, the value of ^ is =( — ) 2 n; 
whence 
fa a 2 a 3 «§(»- D 1 
(_)«.-.) m _l=4{ T ^ +rTsl + iT76 ... + T ^ 1 }, 
a theorem relating to the imaginary nth roots of unity, n an odd prime. In particular 
n= 3, — 4 = 4-j ^ at once verified by a 2 +a + l=0, 
n=5, 4 = 4 {rr^+x~ ij, verified by a 5 -l = 0 
(viz. the theorem is also true for the real root a=l) ; in fact the term in { } is 
(a(l+a 4 ) + a 2 (l +a 2 )^ -r-(l+a 2 )(l + a4 )! that is (a+l+a^'a 4 )'— (l+a 2 + «- 4 -{-a), =1 ’ 
n= 7, -8=4( t ^- 2 + 1 -^- 4 + t ^-6.|, 
ll+a 2 1 l+a* 1 1 +a°J 
which may be verified by means of a 6 +a 5 +a 4 +a 3 +a 2 + a+l = 0 ; and so on. 
55. I further remark that we have 
JL _ f _ p(»- o J sn 2a> 0 , sn 4eu 0 . . sn (n - 1) co 0 \ 
M 0 ' ' (snc2w 0 .s 
.snc 4w 0 . . snc ( n — 1) w 0 
But Jacobi (p. 86), 
where (p. 89) 
that is, 
Hence 
o 2K0 
sn Zgu, =sn > 
AK 1 (l-(fe^) . . (l-y - V 2 *) . . 
m (1 — qe 2i %) . . (1 — qe~ 2i ^) . . ’ 
_ i J~(l + g)(l+? 3 ). ■ .~| 2 _ x-P- 
Vk' 
(l + 2*;(1 + ?«)... 
sn ^ga—j q . 2faff ^—ei^q ) . . (I — «»-' 2 sq ) . . 
sn 2gco 0 _ u?s— 1 1 — a 2 ^ 2 . . 1 + a?Sq . . 1 — u n ~^q z . . 1 + u n ~ 2 sq. . . 
snc 2gao 0 ~ i(u 2 s + 1) 1 + a 2 Sg 2 . . 1— a 2 ^ . . l+a“- 2 *gr 9 . . 1 — a»- 2 *gr. . * * 
and giving to g the values 1, 2, . . . -J {n — l),and multiplying the several expressions, we 
have the value of viz. this is 
ri-r’n^jw 
where K,(g) denotes the product of the several factors which contain q. 
56. The ( i 2 ) of the denominator gives a factor i n ~ l , =( — ) 2 , which destroys the 
factor (— )~. We have then a factor 
n (S) S ’ which is =(-) K ”" >n - 
