ON THE THEORY OF NUMBERS. 



333 



tt*=$*(~\. If we represent the modular equation of order 2*, when 



cleared of fractions, by/(2", u, «v) = 0, the modular equation of order 2' x+1 , 

 qi f(2P +1 , u, ^ + i)=0, is obtained by eliminating v^ from the two equations 



/(2^, «, ^)=0, and ^V+i= i+ J* ' We may thus successively calculate the 



modular equations of the orders 4, 8, 16, ... ; and, attending to the ex- 

 pression, by means of the transcendent 0, of the roots of the equation 



v* = . we may establish the following properties : — the function 



/(2i*, u, Vp) is of the order 2 M ~ 2 in v s , and of the order 2^ in ir ; the co- 

 efficient of u 2 * +1 is t/ 2,l+ \ and the equation is not altered by writing - , for if 



and multiplying by xv^ + ; if w=^(w),the values of v are given by the equation 



* 8 =* 8 (^')' . ( 39 > 



in which h represents any term of a complete system of residues, mod 2'* -2 , and 

 correspond to the transformations defined by the formula 



c+da 



a, b 

 c, d 



1,0 



—81; 2* 



x 



1,0 

 27*, 1 



a 



a+Ml 



where h is any term of a system of residues, mod 8 ; if v=tp{€l), the values 

 of u are given by the equation 



/ 2' x £i \ 



■=*iT+m)> < 4 °) 



where h is any term of a system of residues, mod 2*\ 



For the determination of the multiplier in a transformation of an uneven 

 order n, we have the theorem, 



" If M is the multiplier corresponding to the transformation 



u= 12l2L_, the $(«) quantities s=(— 1) 2 — satisfy an equation of 



y Jil- 



order *(«), in which the coefficient of the highest power of 2 is unity, and 

 the coefficients of the other powers of z are rational and integral functions 

 with integral coefficients of k 2 ; the absolute term, in particular, being +»"*. 

 126. The Complex Multiplication of the Argument. — The problem of the 

 multiplication of the argument is " Given an integral number n, to express 

 the Theta functions of nx and w by means of the Theta functions of 00 and w." 

 The solution of this problem may be made to depend on that of the addition 

 of arguments ; for to add n equal arguments is to multiply the argument by n. 

 The problem is also included in that of transformation ; for if we consider 



the transformation of order » 2 , of which the matrix is A ' . wehave£2=w, 



' U, n Y ' 



A = K, A' = K', i=«- 



When w is not the root of a quadratic equation having integral coefficients, 



* Jacobi in Crelle's Journal, vol. iii. p. 308. M. Joubert (Comptes Eendus, vol. xlvii. 

 p. 341) has calculated the equations of the multiplier for the orders 3, 5, 7, 11. See also 

 M. Brioschi in Tortolini's Annals, vol. i. (New Series) p. 175, M. Hermite, Equations 

 Modulaires, pp. 12 and 31. No complete demonstration of the theorem appears to have 

 been given. 



