356 keport — 1865. 



where 



a, b 

 c, d 



J'J-,mod2j c=0, mod8; (-l^^W-Y The 



determinant of this equation, if a — \(ay + dy') — Skb, is a 2 — n. For brevity, 

 let us suppose that »= — 1, mod 16 ; we shall now prove that in this case 

 o- = 0, mod 8. Since «c7 = l, mod 8, it follows that « = (/, mod 8 ; let 

 a=8a+/u, d=8$+fi; substituting these values in the equation ad— bc=l, 

 considered as a congruence for the modulus 16, -we infer that 8(a + (5) = 

 c + fj. 2 — 1, mod 16. Again, since yy'=— 1, mod 8, let y=89+»', y'=80' — v\ 

 substituting these values in the congruence yy = — 1, mod 16, we find 

 8(0 + 6') = v 2 — 1, mod 16. But 2 ff = cty + dy' = 8[a + a + + 0'] ^ 



c ^—l v-—l e 



c+/i =_l + ^_l=E0, mod 16, because (-1) 5+ 8 ~~=(-l)¥-Y-) 



=1. Therefore a is divisible by 8, and the quantity w is the root of an 

 equation A+2Bui + GV=0, in which A is even, C uneven, and of which 

 the determinant is included in the scries of negative numbers, — n, — n + S 2 , 



— «+16 2 , An application (which we need not here repeat) of the 



method already employed to prove the formida V. will show that every qua- 

 dratic equation satisfying these conditions supplies a value of <j>(w). Sixteen 

 different values of <t>(w) will be obtained from the quadratic equations associated 

 to the forms of any uneven class of a determinant included in the above series ; 

 because the conditions with respect to the extreme coefficients are satisfied in 

 only two of the six subclasses contained in each class, and because each of 

 these two subclasses supplies (art. 126) eight values of f(w). Lastly, the multi- 



f(v) 

 plicity of the root <£(u>) of the equation JV ' — = is ascertained, by an ap- 



(x — 1) 

 plication of the method of M. Joubert (which also we need not here repeat), 



to be equal to the number of factors 1 — ( - WwW ' ya 'Z_ J which are 



annulled by w ; or, which comes to the same thing, to the number of repre- 

 sentations of n by the form a- 2 + At 2 , the first indeterminate being divisible by 

 8, the second being uneven and positive, and — A representing the determi- 

 nant of the primitive equation by which « is determined. Denoting by («, A) 

 the number of such representations, we have for the whole number of roots 



of the equation — ^-2 — =0, each root being taken with its proper multipli- 

 city, the expression 162(;i, A)7^A) ; whence, by a transformation already em- 

 ployed, 



16F00 + 32F(n-S 2 )+32F(>i-16 2 ) + ... 1 ,.. 



= 2*(»)-4[*(H)-¥(n) + *'(»)-*' (»)]•/ ' * * * W 



Considering, instead of the function f(x), the function x^^f l X) _- J , we 



obtain, by reasoning precisely similar, the formula 



32F(n-4 2 ) + 32F(«-12 2 ) + 32F(ji-20 2 ) + . . . \ (m 



= a*(H)-4[*(»)-*(n) + *'(»)— *'(»)]» J ' ' ' ' 

 whence, by subtraction, 



2F(n)-4F(n-4 2 ) + 4F(«-8 2 )-4F(n-12 2 ) + . . . 

 =¥'(n) — $'(«)> 

 in accordance with M. Kronecker s formula VII. 





