400 MAJOR P. A. MACMAHON OX THE THEORY OF PARTITIONS OF NUMBERS. 
Summing similarly the series 
we find 
a. 
'S+t 
+ 0-25+ 2< + 0-:U+3t • • • 
2 (*' + 0 ('J i d“ ^2 — Ji) + +2^ — 0 
+ (i-s + — ’R) ^2 + {I + m — — \t) Js, 
and we have in each of the ten cases to establish the relation 
1 (s + 0 (li + II - Is) + {¥ -¥-i) I. 
+ (■S'5 ~ ~ I 2 + (^ + ~ + ¥) i3> 
> 1 (s + ^) {j\ + _ ji) + + ¥-i) Ji 
"h (2^ “h •I2 d“ d" — 2 '^ —¥) ’I3 
for all values of s and t. 
For Case 1 it reduces to 
I, + I 2 S I:„ 
which is true, for here Ij + 1-2 = T 3 . 
For Case 2, making use of Jj + J 2 = J 3 ~ 1, the reduction is to 
I — s —t 
s + t 
and Ji being the greatest integer in 
I - 1 
, and moreover 5 4 -*^ being at least unitv, 
the relation is obviously satisfied. 
For Case 3, making use of J, + Jo = J 3 , we find 
I — s 
s + t’ 
and this is satisfied as s > 1 , 
For Case 4, reducing by Ji + Jo = J 3 — 1 , we find 
rn. 
J2^ A - 1 
S “I" 2^ 
obviously true from the definition of Jo. 
For Case 5, reducing by Jj + Jo = Jg, we find 
Jo> 
m — s 
s + t 
obviously satisfied. 
For Case 6 , reducing by Jj + Jo =: Jg — 1 , we find 
J3> 
clearly satisfied. 
I + VI — s 
S "f" / 
