OF THE ROOTS OF SYSTEMS OF EQUATIONS. 513 
55. I also give the developed expressions of a few of the partition operators. 
Thus 
.^(To) = 3(70) + (10) 0(IO2) -h (dl) 0(IOO1) + (20) 3(20 10) + (10^) 3(103) 
(11) 3(nio) + (10 01) 3(702 oT) + (02) 3 , 7002 ) + (01^) 3(7ooT2) 
+ . . . , 
^/(20 OT) — 3(20 OT) "T (10) 3(20 To OT) "T (dl) 3(20 OT^) “h (20) 3(202 oT) 
-f- ( 10 ^) 3(20 I 02 ol) "h ( 1 1 ) 3(20 IT Ol) (dd 01 ) 3(20 To 0 T 2 ) 
+ (d2) 3(20 Ol 02) + (01^) 3,2ooT3) + • • • 
The mode of operation of the biweight operators in the separation theory is now 
manifest. 
56. Let 
...)■> i7(ToPw oTP"'... ...) ’ 
be any two partition operators of the same or different biweights. Representing 
them for brevity by 
ff (»r) 5 f/(p) ’ 
we have 
9m9(.p) 9m9(.p) ”h 9m “j“ 9(p)’ 
wherein the multiplication on the left denotes successive operation, the bar on the 
right denotes symbolical multiplication, and the symbol "j" denotes explicit differentia¬ 
tion on the operand regarded as a function of symbols of quantity only. 
It is easy to establish the result 
9M "j" 9(p) — 5^(10'^“ + P'O OI’^O' + Poi . .. + Pmi . . .) 
= 9{- + p) foj' brevity. 
Hence 
9m "1“ 9(p) = 9ip) "j" 9m = 9in + p), 
and 
or at full length 
9M9ip) — 9 m9{p) + f/U + p); 
^(lo’^io Ol’^oi . . . 2,12^^121 , . ,) f/(lo'’l'> OlPoi . . . . . .) — 5^(T0^io Ol’^oi . . . Piq^Pih . . .) 5^(10^10 oFm . . . . . .}.' 
+ P'dO^io + Pio oi’Toi + Poi. . , . . .), 
the fundamental law of multiplication (compare Art. 15). 
MDCCCCX.—A. 3 U 
