OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
519 
produces the same result identically. This result is composed of three portions 
containing separations of the partitions 
(I^’Tio-s 20^20 30’"3o _ _ (10’"““^ 20’"““^ 30’'“ . . .), (10’'“ 20"“ 30"““^ . . .) 
respectively. Hence the operations which produce the three identically equal portions 
of the result must be equivalent, and the three relations between the operators there¬ 
fore follow. 
In general, we may say that of the biweight there are as many relations between 
partition operations as there are partihions of the biweight pq. 
65 . The general law of the coefficients will be now investigated. 
In the result of Art. 53 , viz., 
we have to substitute for G^^^^ . . ., the sums of the partition G operators of 
weights Piqi, p^q^ • • •, respectively; we have then to collect on the right all the G 
products which are associated with separations of one and the same partition, and to 
equate them to the corresponding g operator on the left. It is evident that this 
process does not alter the laiv of the coefficients, and that representing the different 
separates of the given partition by (J^), (J^) . . ., and any separation by (Jg)'^" • • •> 
we may write 
- 1)! .. 
) „ I _ ! .V 
TTj! 
I Y./ — 1 )' Ji 
Jl- J 2 - • ■ • 
Observe that this is precisely the law which gives the expression of a single bipart 
function in terms of separations of the paidition (^>1^1"' p>i^'2^ . . .). I recall the result 
of Art. 41 , viz.. 
(-) 
277 
_i( 27 r - 1)! 
TTi! TTj 
CMi" 
= ^A-) 
- 1)! 
7 i!A- • • ■ 
which renders the correspondence between the algebraic and differential theories very 
striking. 
66 . Reversing the formula as in the algebraic theory we get the important 
formula:— 
(Stti — 1) ! (STTg — 1) ! . ■ . 
yd/d . . . TTidT^ia! • • • WgdTTad 
ii H 
r 12212"“ • • •) 9 (p 2 i 22 i"^^ E2W® • • •)• 
