OF ELLIPTIC FUNCTIONS. 
419 
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 = \ or 7 (mod. 8) the sign is +. And there is in every case a pair 
of terms v n 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 n— 19, the lines parallel to the dexter dia- 
gonal are —1 (symmetrical), +114, —114 (skeAv), 0, —2584, —6859, —2584, 0 (sym- 
metrical), and so on. The same relation of symmetry is seen in the composite cases 
n — 9 and n= 15, both belonging to n = l 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 — 1)“ +1 = 0, but in the case n = 3 or 5 (mod. 8) 
it becomes (v-f l)”^— 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 5-th root of unity, a value of v is 
±v 
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=lh, the values are 
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 3, then we have 
04 + 20V-2^-m 4 =O; 
and to each of the four values of S corresponds the six values of v belonging to the 
quintic transformation given by 
4^505 _|_ 5^2 _ £^ 204 _ 4^0 _ ^6 — 0 . 
The equation in v is thus 
. . -Q<i)(v 6 . .* -0l)(v 6 . . -0t)(v 6 . . -0®)=O, 
where 0 1? 0 2 , 0 3 , are the roots of the equation in Q. viz. we have 
d i +2(Pu 3 —2Qu—u i =(0—O l )(O—() 2 )(()—d 3 )(0—0 i ) ; 
