514 
MAJOR MACMAHOR ON SYMMETRIC FUNCTIONS 
Also 
9m9{p) 9{p)9u) ^5 
or 
9{W^o . . . p^Pi2i . . ,) ,9 ^(To'’'» OlPol. . . . . .) OlP»i . . . ■E5 I'’Pi5i . . .) 5^(To"io oUoi . . . . ..) — 0, 
shewing that any two partition operators are commutative. 
57. The left hand side of the result just reached is called by Sophus Lie"' the 
“ Zusammensetzung ” or “ Combination ” of the operators which appear. Sylvester! 
has also called it the ‘ Alternant’ of the two operators. 
The whole system of partition operators forms an infinite group in co-relation with 
an infinite group of transformations— 
We can state the theorem :— 
Theorem. 
“The Combination or Alternant of any two partition operators vanishes.” 
Considering the partial differential equation 
9m — 
and <f) any function which is a solution, then must g{p)(f) be also a solution, since 
9{^) 9(p) - 9ip) 9 m </> = 0 . 
Theorem. 
“ If and p(p) be any two partition operators, and (/> a solution of the equation 
P(,,) = 0 ; then will p(p) ^ be also a solution of the same equation. ” 
58. Consider now the partial differential equation of Art. 51, 
(-) 
p + <1 
_i (jy+g-Jd ! 
p! q\ 
9pq 
= Y 
(-W-i(^7r-l): 
77 ! 77 
77 
9^ 
Pill- 
(To^io . . . 'p^Vih . . .) 
= 0. 
Assume the separable partition to be 
so that the operand is a linear function of separations of this partition. 
The effect of the partition operator 
is the production of terms each of which is a separation of the partition 
( 10 *^“ ~ 01 “^“^ “ ~ ...). 
* ‘ Theorie der Transformationsgruppen,’ Leipzig, 1888. 
t ‘ Lectures ou the Theory of Reciprocants.’ (‘American Journal of Mathematics,’ and elsewhere.) 
