778 REPORT—1863. 
®, (a2 = 
eye ie Rta Avan 
since 
_24 ae 4S 
Ap &Sy™+4Bp AEy™ == —(—1) 9 ahaa mod p, 
za za 
Ap 4¥y"4Bp 4zy”= 0, mod p, whence by addition 
> 4 
“= —(—1)a——__—_., mod p. 
= A tertian 
If Ais a prime, x also satisfies the congruence }v =1, mod A; for the 
sum of the coefficients in any function y (m, n) is== —1, mod p—1, and 
therefore mod A; whence the sum of coefficients in y (@) which is a product 
of an even number of such functions is 1, mod A; because the reduction 
of x (0)-to the form AZ6*”+ Bre” is effected only by means of the equations 
J gee es 
Q” 
gr4_]—0, =(0; whereof the former does not alter the sum of the 
coefficients at all, and the latter alters them only by a multiple of A. Con- 
3 
sequently + (A—1) (A+B)=1, mod A, or, since p&X =1, mod A, and 
(—1)=—1, 4v=1, mod A. 
It will be observed that if A is of the form 8m-+ 7, whether A is a prime 
or not, # and y are necessarily even in the equation (A); whence, dividing 
by 4, we may put the equation in the form 
sb— Sa 
P A =0" + Ay? 
3b 
ie ed 
soa Ty ayes TI Man’ ube 
Ex. Let A=7, p=7n+1; the values of aare1,2,4; of b,3,5,6: hence 
ik ae [13x 
— 2 Tin. T12n 
potty, «= —} Sa ,mod p; alsow==1, mod 7, 
(Jacobi, Crelle, vol. ii. p. 69.) 
Whenever the exponent of p is 1, the formule (A) and (B) completely de- 
termine the value of #; when the exponent of p is 2, we can only be sure 
A-1 
expression = (5) f («)==f (®*) —=/f (0), the equivalent infinite series 
a 
A m=o A Ls 
ey tT () det 23 i} cos lh lel 9 ds; 
4 Jo Am=1 J0 
whence, observing that 
z=A—1] ee e=A4—-1 1, Shiny 
= (Zain -(*) VA, and = (2) cos —— =0, 
e=1 a a s ; m=A Z 4 
; m= 8 ‘A . 
we obtain $ VA (f(a) —f (t))= = : @)(. sin cal ds; a formula from which 
m= 
Dirichlet’s result is immediately deducible, by putting f (7) =«, and performing the inte- 
grations. It is a remarkable fact that the inequality 2b > =a has never been proved by 
elementary considerations, or without the use of infinite series (see the Memoir on the Arith- 
metical Progression, p. 57). If A is a prime, 2}—2a is certainly not zero, for 2b+4- a is 
uneven (because A is of the form 4u+3); but even this remark cannot be extended to the 
case in which A is composite. 
* 
SWS AS 
