776 REPORT—1863. 
Sb— zu 
ip A =w + Ay? 
3b 
8 aki i br 
ogi “TL, [Nan] 
In these formule the signs of summation extend to every value of a and 6 
respectively ; and in the expression IT, [flan] the exterior sign of multiplica- 
tion II, extends to every value of a, while the interior sign is the factorial 
symbol, so that Wan=1.2.3...an. The number 3 is excluded from the , 
first formula; the numbers 1 and 2 from the second. 
It will suffice to show how the first of these two theorems is to be demon- 
strated. For this purpose we consider the product IF (0°); taking an, 
a'n, a'n, .. for m,, m,',... we find y (0)= — TF (6%); because (as may 
easily be proved) Sa==0, mod A, whence San =0, mod p—l. We shall 
now show that (0) is of the form AZ0“"4+Bx0"". Actually multiplying the 
expressions F (6~“”"), F (0-*"),...., the coefficient, in the product, of any 
term such as v* @”” is equal to the number N of the solutions of the simul- 
taneous congruences 
Pty ty" +..=k, mod p, ay+a'y'+a'y"+....== —m, mod A. 
If r is a number prime to A, and satisfying the equation = =+1, N will 
not be changed, if we write rm, ra, ra’, . . . (or rather the least positive residues 
of those numbers, mod A) for m, a, a’. Hence, in y (@) all powers of 0 whose 
exponents are of the form an have the same coefficient A’, and all powers of 0 
whose exponents are of the form bn have the same coefficient B’. Again, con- 
sider a power of 0 of which the exponent is of the form aén; 6 representing a 
given divisor of A (other than 1 or A), and « representing any number less than 
> and prime to . ; all such powers of 6 will have the same coefficient. For 
we can always find a number r prime to A, satisfying the equation (5)= us 
A f By 
and yet congruous, for the modulus 5? to any given number prime to 33 
whence it follows that the number N will remain the same for all values of 
m included in the formula a3. But a sum of the form 3,,0%” is equal to+1 
or—1, according as the number of primes dividing 5 is even or uneven, be- 
A 
cause it is the sum of the primitive roots of the equation w'=1. Thus, the 
function y (4) assumes the form A'S6” + B's0°"+C', whence, attending to the 
equation £44 30°"=(—1)', in which ) is the number of primes dividing 
A, we find, as has been said, —IIF (0-*")= x (0)=AZ0"+Bz0". If we 
write @~' for @ in this equation, it becomes 
—IIF (6%")=y (071) =Az0"" + BEO™, since (=)= i (=)= ce i 
Multiplying the two equations together, and obserying that 
F (6-”) F(6")=(— 1)" p=p, because n is even, we obtain 
4pY¥o) —[(—1) (A+B)+(A—B) 2 ia | 
x [(—1)* (A+B)+(A—B) (26"—=6")], 
| 
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4 
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