ON THE THEORY OF NUMBERS. 335 



into the other, is or is not of the type 



mod 2. We have therefore 



1,0 



0,1 



only to ascertain how many subclasses each class contains, a subclass con- 



1,0 



sisting of forms equivalent by transformations of the type 



0,1 



A simple 



discussion shows that the number of subclasses is six (corresponding to the 

 six types of binary matrices for the modulus 2) ; except in the two cases just 

 referred to, when the number of subclasses is reduced to 3 and 2 respectively, 

 owing to the existence in those two cases of automorphics which are not of 



, mod 2. Thus the whole number of values of ^ 8 (w) is 6G(A), 



the type 



0,1 



G(A) representing the number of classes of det. — A, counted in the manner 

 stated above*. It will be seen that the six values of <p a (w) corresponding to the 



forms of the same class are of the type k 2 , — , 



(being in fact related to one another as the six anharmonic ratios of four 

 points). The three values corresponding to the forms of det. — 1 are — 1, 2, i ; 

 and the two values corresponding to the improperly primitive forms of det. 

 — 3 are the imaginary cube roots of — 1. 



It is an important theorem (to which we shall again refer) that the 6G(A) 

 values of <£ 8 (aj) satisfy an equation of that order, of which the coefficients are 

 integral numbers (but the first coefficient not, in general, unity). 



The whole number of values of <p(u), corresponding to the forms of deter- 

 minant —A, is 48G(A). For if a be the value of 0(w) corresponding to any 

 form of a given subclass, and -q be any eighth root of unity, r\a will be a 

 value of ^»(w) corresponding to another form of the same subclass. 



127. Jacob? s Formula for the number of decompositions of a number into 

 squares. — The first applications of elliptic formulae to the theory of numbers 

 were made by Jacobi. The developments, in series proceeding by powers of 

 q, of the squares, fourth, sixth, and eighth powers of the functions 



v/ 



— = 1 + 2q + 2q* + 2q 9 + 2 q xs + 



7T 



^5=2 2 i + 2 2 f +2 2 ^+..., 



■K 



which are found in the ' Pundamenta Nova ' (sections 40-42, and 65, 66), 

 are the analytical expression of arithmetical propositions relating to the com- 

 position of numbers by the addition of two, four, six, and eight squares. In 

 these developments n represents any number from 1 to oo, v any uneven num- 

 ber from 1 to go ; d is any divisor of n, S any uneven divisor of n or any di- 

 visor of v ; d' and 3' are the divisors conjugate to d and § ; and the summa- 

 tions indicated by 2„, 2„, 2 d , and 2 extend to every value of n, v, d, and S 

 respectively. 



2K 

 (1) ~ = l+42,(-l) 2 -X_ =1 + 42 « 



i+r 



= 1 + 42 2«(-l) 



* G (A) is the sum of the densities of the classes of det. — A ; the density of a class, 

 according to the definition of Eisenstein, being the reciprocal of the number of its auto- 

 morphics. 



