MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OP NUMBERS. 389 
and investigate the sum 
. . . X“‘ 
for all values of a,, a^, a^, ... a„ which render the expression under the sign of 
summation expressible in a finite integral form for all values of the integer n. 
Art. 106. Let be that factor of 1 — which, when equated to zero, yields all the 
primitive roots of the equation 
1 — = 0 . 
Then 1 — a;* = ifdfd, • • • where 1, f, d^, . . . t are all the divisors of We 
must find the circumstances under which every expression will occur at least as 
often in the numerator as in the denominator. We need not attend to since it 
occurs with equal frequency in numerator and denominator. In regard to we 
have equal frequency if n + I be uneven, but if n + 1 be even we must have 
~{~ “3 “b 0^5 + . . . ^ a, -}- a 4 fi- ag -}- . . . 
For if n + 1 = 0 mod 3, 
“1 “b ^4 “b ^^7 ~1~ • • • — 0^3 “b “s "b “9 ~b • • ‘j 
and if 71 + 1 = 1 mod 3, 
“2 + “5 + + • • • — «3 "b + <^9 + • • •> 
while the case of 7i + 1=2 mod 3 need not be attended to. 
Proceeding in this manner we find the following conditions 
“1 “b “3 + “3 + • • • — “2 “4 "b “6 + 
+ “4 + “7 + 
°^2 + “5 + ^8 + 
r “i + cts + “9 + 
< a 2 + ag + «io + 
®3 + «7 + «ii + 
— “3 4" «6 + «9 + 
> ttg + ag + «9 + 
— “4 + “s ~b “12 + 
— “4 + “3 + ”b 
— “4 + “a + «i 2 + 
r®i “b “b • • • 
1 
. . . . > 
a.,-1 
1 a. 
a,_i 
• 
j • 
« 
- 
r “1 
> 
a. 
j an 
^ .■ 
t 
> 
a. 
1 • 
La,-i 
> 
a. 
\s{s — 1) in number. 
