104 REPORT 1870. 



we shall have all the possible values of this expression. The values of y may 

 therefore be represented by the following expression : — 



>=fi'' \ sii 



4^n'K + 4m'iK' . SmK + Swi'i'K' 



v=n^ \ sm coam z sin coam 



sin coam 



n n 



2(n-l)m'K+2(n-l)m'iK' \ 



n 



1- 



where m' signifies one of the quantities 0, 1, 2, 3 . . . . (n—1), and m is 

 unity, except when m' = l, when it is both unity and zero. 



Wo immediately deduce from the ' Fundamenta Nova ' the equation 



2K.^' 2 J T- l + 2o2rcos2a;+a4'- 



■ Bin coam-^=-, '^q cos ^.-n^ _^_2^,,_, ^^^ 2x+q^r-r 



From this Sohnke proves that the (n + 1) values of (r) may be derived from 



.o ./-r a+r)(l + g^)(l + r/) 1 



(l + 2)(l + f/)(l+'z') J 



by substituting for j successively in this equation the (n+1) quantities 



,= s/2^7i[^ 



111 1 



2", 2"j "S'^j *°2" a" -^2", 



1 is an expression representing all even numbers, the sign being ncg- 



a, being any of the n roots of unity. 



The proof, although long, presents no particular difficulty, and depends on 

 transforming the factors in the continued products by means of the theorem 

 that 2nr+4:in'p (when n is a prime number, wi' one of the numbers 1, 2, 3 



n — 1, r all numbers from zero to infinity, p all numbers from to 



n—l 



~2~J 



looted. 



Section 3. — It appears from this investigation : 



A. That the modular equation is of the (n+l)th degree. 



B. That the coefficients of the equation, when arranged in powers of v, 

 are rational and entire functions of fx. 



C. That the last term of the modular equation is of the form +^"+' if 

 (n) is of the form Sr + l, — /x"+i if n is of the form 8r + 3. 



This is deduced by Sohnke from the observation that it is a consequence 

 of the multiplication of elliptic fimctions that all the roots should have the 

 same sign as the quantity 



ii" i 



4K . 8K . 2(n-l)Kl 



sm coam — sm coam — . . . . sm coam ■ — > . 



n n n ) 



D. The modular equation is unchanged when Ic and \ are interchanged, 

 therefore the highest power of (/i) cannot exceed (n+1). 



E. We have already seen that n is of the form \/2^qf{q). One value 

 of {y) must therefore be of the form \/2'{/q^'f(q"). If we substitute this in 

 the modular equation, the irrationality must disappear. Hence in any term 

 of the modular equation aj.i'")'^, we must have ^q'"' . ^ q™ = 2' \/(l*> where 

 t is constant for every term. Hence m+rnz=Ss+t, and therefore the mo- 

 dular equation must be made up of terms of the form 



j''-(a/xP + /3/xP+8 + y;uP+lH ). 



