MAJOR P. A. MACMAHON 05^ THE THEORY OP PARTITION’S OF NUMBERS. 399 
Case 4. 
Il = Jl 
• 
Ii 4 ~ I2 — I3 
I2 — J2 4 * 1 
T T 
-•-3 *^3 
Case 5 . 
Ji 4 “ J2 = J3 — 1 
Ti = J, 
Ii 4 " I2 = I3 4 " t 
] 2 = J.3 4 - I 
I3 — J3 
Case 6. 
J, 4- J o = J, 
Ti = Ji 4 “ 1 
Ii 4 " h’ = I3 4 “ i 
I, = Jo 4 “ 1 
I3 — '^3 
Case 7 . 
J1 4 “ J2 — J 3 — 1 
1—1 
’ll 
-h 
Ii + I2 = I3 
L = J2 
J, 4- ,Jo = J 3 
I3 = J3 4 " 1 
Case 8. 
Ii = Ji 
I) 4 " l2 = I3 
12 — J2 4 " 1 
13 = J3 4 “ 1 
Case 9 . 
J1 4 “ J2 — J3 
Ii = Ji 4 ~ 1 
Ii 4 - I2 = I3 
12 = J2 4 “ 1- 
13 — J3 4" 1 
Case 10. 
.T J 4 ~ J 2 — 4 3 t 
Ii = Ji 4 " i 
Ii 4 “ I2 = I3 4 “ t 
12 = J2 4 “ 1 
13 == J3 4 " 1 
J1 4 “ J2 — J 3 
For the series 
“s + “2s + / + “3 j + 2« + • • • 5 
we have, as far as a,_i, Ii terras ; as far as Ig terms ; and, as far as + I 3 
terms. 
Hence the summation gives : — 
i'll { 2 s + (Ii — 1 ) (s + ^)} H- ? (I 2 — Ii) 
“h i" (I 3 — I 2 ) {2^ 4" ‘ 2,171 -j- 2 ^ — 2 (s -4“ (I 2 4" 1) “ (s 4“ 0 (^3 ” ^2 — 1)} 
= i (5 4- 0 (Ii + I 2 — 10 4 - "* 2 ^ ~ iO 
4 - ( 2 ^ — — m) I 2 4- 4" — 2 -^ 4- aO 
