136 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578 
Various remarks arise on the Tables. Attending first to the cases n a prime 
number ; the only terms of the order n +1 in v or u are v n+l + u n+1 , viz. n = 3 or 
5 (mod. 8) the sign is —, but n = 1 or 7 (mod. 8) the sign is +. And there is in 
every case a pair of terms v il u n and vu, having coefficients equal in absolute magnitude, 
but of opposite signs, or of the same sign, in the two cases respectively. 
Each Table is symmetrical in regard to its two diagonals respectively, so that 
every non-diagonal coefficient occurs (with or without reversal of sign) 4 times ; viz. 
in the case n= 1 or 7 (mod. 8) this is a perfect symmetry, without reversal of sign; 
but in the case n = 3 or 5 (mod. 8) it is, as regards the lines parallel to either diagonal, 
and in regard to the other diagonal, alternately a perfect symmetry without reversal 
of sign and a skew symmetry with reversal. Thus in the case = 19, the lines parallel 
to the dexter diagonal are —1 (symmetrical), +114, —114 (skew), 0, — 2584, —6859, 
— 2584, 0 (symmetrical), and so on. The same relation of symmetry is seen in the 
composite cases n = 9 and n= 15, both belonging to weI or 7, mod. 8. 
If, as before, n is prime, then putting in the modular equation u = 1, the equation 
in the case n = 1 or 7 (mod. 8) becomes (v — l) n+1 = 0, but in the case n= 3 or 5 
(mod. 8) it becomes (v + l) n (v — 1) = 0. 
27. In the case n a composite number not containing any square factor, then 
dividing n in every possible way into two factors n = ab (including the divisions n. 1 
and 1. n), and denoting by /3 an imaginary 6th root of unity, a value of v is 
± \/2(/3q b Yf(Pq b )\ 
so that the whole number of roots (or order of the modular equation) is = v, if v be 
the sum of the divisors of n. Thus n = 15, the values are 
V2q'^fiq 15 ), -V2g***/(?*)» - ^2g***/(?*), V2?*•*/(?*) 
1 , 3 , 5,15 roots ; 
and the order of the modular equation is = 24. The modular equation might thus be 
obtained as for a prime number ; but it is easier to decompose n into its prime 
factors, and consider the transformation as compounded of transformations of these 
prime orders. Thus n = 15, the transformation is compounded of a cubic and a quintic 
one. If the v of the cubic transformation be denoted by 0, then we have 
04 + 203 u s _ 2da -u 4 = 0; 
and to each of the four values of 0 corresponds the six values of v belonging to the 
quintic transformation given by 
v 6 + 4iv 5 0 s + 5v 4 0- — ov 2 0 i — 4 v0 — 0 fi = 0. 
The equation in v is thus 
(<v 6 + év s 0! s + .. - 0 1 6 ) (if + .. - 0«) (v 6 + ..~ 0 3 e ) (v 6 + .. - 0 4 6 ) = 0,