functions. [578 
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 127 
k- + Sk + 1 = 0, or else 
uation in k is 
ion on putting therein 
l) 4 } at least twice, and 
l) 6 } belonging to the 
A l) 4 ] presents itself 
on further. 
lifficulty of the purely 
instance the modular 
with special factors of 
taining the results in 
op the theory, repro- 
lations, and extending 
of q, viz. 
■h-) 
itities in question are 
d unless the contrary 
transformations corre- 
f be distinguished as 
unity, we have 
n ff(ci-q n ), &c., 
The occurrence of the fractional exponent ^ is, as will appear, a circumstance of 
great importance ; and it will be convenient to introduce the term “octicity,” viz. an 
expression of the form q**F(q) (f= 0, or a positive integer not exceeding 7, F (ç) a 
rational function of q) may be said to be of the octicity /. 
24. The modular equation is of course 
say this is 
(v-v 0 ) (v-v 1 )....(v-v n )=0; 
v n+1 - Av il + Bv n ~ l - ... = 0, 
so that ^l=Sw 0 , B = 1v 0 vj, &c. In the development of these expressions, the terms 
having a fractional exponent, with denominator n, would disappear of themselves, as in 
volving symmetrically the several nth roots of unity ; and each coefficient would be of the 
11 
form q H F(q), F a rational and integral function of q. It is moreover easy to see that, 
for the several coefficients A, B, G, , g will denote the positive residue (mod. 8) of 
n, 2n, 3n,... respectively. 
Hence assuming, as the fact is, that these coefficients are severally rational and 
integral functions of q, it follows that the form is 
au° + bu g+8 + cu g+16 + ...., 
g having the foregoing values for the several coefficients respectively. And it being 
known that the modular equation is as regards u of the order —n+1, there is a 
known limit to the number of terms in the several coefficients respectively. We have 
thus for each coefficient an identity of the form 
A = aiiP + bvP +i 
where A and u being each of them given in terms of q, the values of the numerical 
coefficients a, b,.. can be determined ; and we thus arrive at the modular equation. 
25. It is in effect in this manner that the modular equations are calculated in 
Sohnke’s Memoir. Various relations of symmetry in regard to (u, v) and other known 
properties of the modular equation are made use of in order to reduce the numbei of 
the unknown coefficients to a minimum; and (what in practice is obviously an impoitant 
simplification) instead of the coefficients 1v 0 , &c., it is the sums of poweis Sv 0 , 2r 0 , 
&c., which are compared with their expressions in terms of u, in order to the deter 
mination of the unknown numerical coefficients a, b,... The process is a laborious one 
(although less so than perhaps might beforehand ha\e been imagined), involving veiy 
high numbers ; it requires the development up to high powers of q, of the high powers 
of the before-mentioned function f (q) ; and Sohnke gives a valuable Table, w ic 
reproduce here ; adding to it the three columns which relate to $q.