ON THE THEORY OF NUMBERS. 351 



/ 8 (#,1— a?)4-«*W+*<^ 



is finite, when x increases without limit, or f 6 (x, 1— x) is of the order 

 #(»)+¥(n). 



(ii.) Neither nor 1 is aroot of the equation / 8 (.v, 1— x)=0; for/ 8 (0, 1)=1, 

 / s (l, 0) = 1 ; therefore (art. 125) any one of its roots can he represented by^ 8 (w), 

 to denoting an imaginary quantity, in which the coefficient of i is different from 



zero and positive. But if x=f(u>), f s (x, 1—x) = n |V( W )— <p* /y M + 2 ^\"l ; 

 hence the supposition that <j> s (w) is a root of the equation/ 8 (#, 1— .r) = im- 

 plies that \p\w)=<j> a / y fa) + - J \ f or one system (at least) of values of y, y, 



and Tc ; i. e. (art. 125) that there exists an unit matrix 

 the" equation 



a,b 

 c,d 



satisfying 



yw + 2k_C + (1w ... 



7 



and the congruence 



a, b 

 c, d 



, mod 2 (A') 



0,1 

 1,0 



Thus, if <t> 9 (w) is a root of/ 8 (a', 1 — x)=0, w is the root of a quadratic equa- 

 tion 



2al-—cy' + 2{bTc — ±dy' + §ay)w 4- 6yu> 2 = 0, 

 whose extreme coefficients are both uneven, and whose determinant, if 

 ff = — bh-\-^(ciy-\-dy'), is <r 2 — n, a number necessarily negative, because w is 

 imaginary. Conversely, if u» is the root of a quadratic equation, of which 

 the extreme coefficients are both uneven, and of which the determinant is 

 negative and included in the formula a 1 — n, <t> s (w) is a rootof/ 8 (;r, 1— ,r) = 0. 

 Or, more precisely, if w is the root of a properly primitive quadratic equation, 

 of which the determinant — A is negative and the extreme coefficients are both 

 uneven, and if n can be represented by the form (1, 0, A) with a positive 

 and uneven value of the second indeterminate, </> 8 (w) will be a root of 

 / 8 (.r,l — #)=0, and the multiplicity of this root will he equal to the number 

 of such representations of »*. To establish this, we shall show (a) that w 



annuls as many of the factors ^(w) — f 9 1 - — -, — I as there are representa- 

 tions of n; (/3) that f s (x, 1— x~) is divisible by x — <p s (w) as often as there 

 are factors annulled by w. (a) Let A+2~Bo> + Cto 2 —0 be the equation satisfied 

 by w. and let n = a 2 + Ar 2 ; A, C, and r being positive and uneven ; the four 

 equations 



by = T C, -bJc + i(ay + dy') = a } \ ' ' W 



will supply one and only one system of values for y, y', and Tc, and one and only 



one unit-matrix ' , satisfying the equation (A) and congruence (A'). For 



the equations ay—T~B + <r, 6y=rC, show, that y is the greatest common divisor 

 of rB + o- and rC ; this common divisor is a divisor of n , because 

 ft = <r 2 +Ar 2 = (<r-Br)(cr + Br)4-ACr 2 ; 



thus a, o, y, and y'=- are determined. Again, the congruences 

 7 

 * The method by which the multiplicity of the roots of the equation /„(#, l-#) = is 

 here determined is chiefly taken from M. joubert's Memoir, " Sur la Theorie des Fonctions 

 Ellipriques," &c, pp. 22-24. 



