410 
PEOFESSOE CATLET ON THE TE AN SEOEMATION 
viz. writing a to denote an imaginary n-t\i root of unity, we have 
».= _ %/2 /'/(A 
(Observe (— ) 8 =+ for w=8p + l, — for n=8p+ 3.) 
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 expres- 
sion of the form q* F(q) (f= 0, or a positive integer not exceeding 7, F(g') a rational 
function of q) may be said to be of the octicity f. 
24. The modular equation is of course 
(v-v^iv-v ,) .... («— ®,)=0 ; 
say this is 
v n+1 — Au ra +Bfl“ -1 — . . . =0, 
so that A=£v 0 , B= 2^1, &c. In the development of these expressions, the terms having 
a fractional exponent denominator n would disappear of themselves, as involving symme- 
g 
trically the several n- th roots of unity, and each coefficient would be of the form q a F(q), 
F a rational and integral function of q. It is moreover easy to see that, for the several co- 
efficients A, B, C, g will denote the positive residue (mod. 8) of n, 2 n, 3 n , . . . 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 e+8 +cu e+l6 + .... , 
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 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 =au s +bu s * 8 + 
where A and u being each of them given in terms of q , the values of the numerical 
coefficients a, 5, . . 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 number of 
the unknown coefficients to a minimum ; and (what in practice is obviously an important 
simplification) instead of the coefficients Sv 0 , S-y^, &c., it is the sums of powers Sy 0 , Sec. 
which are compared with their expressions in terms of u, in order to the determination of 
the unknown numerical coefficients a, b . . The process is a laborious one (although 
less so than perhaps might beforehand have been imagined), involving very 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, which I reproduce here ; 
adding to it the three columns which relate to <pg. 
