848 
MAJOR P. A. MACMAHOR" OX THE THEORY 
As an example we can find another series for F 
Since 
>1 - lA I (ih - 1 
Fw=i+r 1 + 2 +■••+! 
wherein 
we find 
is the value of f r), 
F (pipa) = 2 "^ * {2h + 2 ) + 2 "^-* - 1) 
, 2p,-i (ih + 1) {p^ + 2) (pa + 6) 
3! 
+ • • 
• J 
a series which it is easy to identify with that previously given (Art. 12). 
15. Useful formula of verification are obtainable. 
It has been shown in Art. 13 that the operation ( —is equivalent to when 
the operand is a function of H only. 
Now 
(a + /3 + . . . — 1) ! 
a 
/S!. 
D “ D ^ 
• 5 
the summation having reference to all positive integer solutions of the equation, 
ot 2,S "1“ 3y . — jji. 
Operating on the expression 
^/{PiPzPs . . r). {jhPilh • • •) 
with ( — d^ for successive positive integral v^alnes of p, and equating the coefficients 
of the symmetric function {x>iJJ 2 P 2 , • > •) find the relations— 
f{P\PiPz •••!,’’) 
= — [fiPiViPi . . . U, r) - 2 /(pj 92P3 ... 2, r)} 
= + {/{PiPzPi . . . U, r) - SfiPiP.Pi ... 21, 7-) + SfiPiPiPs . . . 3, r)} 
_ _j (« + ^ + . ■ . — 1)! s 
«! yS! . . . 
/{PiPiPi • • • 2U% 7’), 
the condition of summation being 
CL -f- 2/3 -f" 3y -f-. . . — s. 
