ON THE THEORY OF NUMBERS. 357 



If we had supposed n=7, mod 16, the left-hand members of the formula? 

 (A) and (B) would have been interchanged, and the right-hand member of 

 the formida resulting from them by subtraction woidd consequently become 



133. "We shall only indicate the origin of the remaining formulas. Of these, 

 the formula I. requires the simultaneous consideration of the modular equations 

 f(2», u\ v e )=0, and /„(», l-« 8 , i< 8 ) =0 (art. 125). Writing x for v°, and 

 eliminating \C~ dialytically from these two equations, we obtain a resultant 

 K(.v) of the order 2 lx+1 (S?(n) in x, as appears from the theory of elimination. 

 "Writing <£*(<«) for x, and observing that the coefficient of the highest power 

 of u in/(« 2 , v s ) is v 2,l+l , and in/ 8 (l — u s , v s ) is unity, we find 



E ( .,)=[»-M/«-n [♦• (j^-)-*-^)], 



an equation which expresses the resultant in terms of the roots of the two 

 equations, and in which the sign of multiplication n extends to every combi- 

 nation of two roots. Since all the roots of / 8 (1— tt* 1)=0 are zero, while 

 none of the roots of f(ir, 1) are zero, no root of the equation E(.v)=0 is a 

 positive unit. But the equations f^l — if, 0) = 0, f(u 2 , 0)=0 have common 

 roots ; so that x = is a root of ~R(x) = 0. To determine its multiplicity, write 

 ai for to in the expression of l\(x), and increase a without limit. The quan- 

 tity^- *^" ) which occurs in *(n) of the factors of B(,r) is equal to 

 \1 -\- 2 (0/ 



f-z^oi), and therefore increases without limit ; but since lim [$ s (ffi)]^ x 



<p -8(2'V)]= 1 > these *( n ) Actors are cancelled by the initial factor 



[VO0] 2 ' 1 * ( " ) - Evaluatm » the remaining factors by the method of art. 131, 



R(.r) 

 we find that lim 24(2>4 -2 . 2 ^ (2M -2 W) is nni te when x diminishes without 



limit; so that the order of R(.r), after division by the highest power of x 

 contained in it, is 2^ +I $ (n) — 2 d> (2> J - 2 n) + 2*(2> i - 2 n), or, since 

 (2*- » - l)*( n ) = *(2^~ 2 ») , 4 $ ( n ) + 2 *(2' i - 2 n) + 2^(2^~ 2 n). 



The formulae II. and III. are obtained by successively combining with 

 the equation / 4 (n, u\ i/*)=0 (art. 125), the equations 



t/V+t^ + u 4 — 1 = 0, and vW -«*+«*+ 1=0, 



the first of which is equivalent to the system v 4 = f 4 ( w ), m 4 =^ 4 /'^V the 



second to the system v i = < j ) \ ( o), K l = — ^ 4 ('£). The resultant of the elimi- 

 nation of tt 4 is, in each case, an equation of the order 2$(«) in x=v i , and is 

 not divisible by x, x — 1, or x-\-l. 



The following Table indicates the highest powers of x and of the divisors 

 of x*—l by which the functions specified in it are divisible. 



