410 REPORT—1883. 
it is shown that a,*-“((u™. y”. . . .)) represents the terms of highest degree in 
symmetric function (A. p™.y”. ... .); so that every known symmetric function 
gives rise to the terms of highest degree in an infinite number of higher symmetric 
functions, by simply making a unit-increase of suffixes throughout (a, having been 
introduced for the sake of homogeneity) and introducing a factor a, raised to the 
necessary power. 
A simplification in the performance of Mr. Hammond’s operator d, is thence 
derived. 
§ 2. It is shown, from analogy with the theory of seminvariants, that the non- 
unitary portion of a symmetric function may be separately calculated, and its 
complete expression thence derived from a series of functions corresponding to the 
coefficients of Prof. Cayley’s generalised canonical form of quantics. 
§ 3. Beginning with Mr. Hammond’s operators, defined by the equation 
d d 
d 
dy =— + @)——= + a= + one 
day aay “dayi9 
the following formula is proved, viz. 
d 
da, 8 wba t Hote cece $(—) "Hedy t . 0. 
a 
wherein H,, represents the sum of the homogeneous symmetric functions of 
weight 7. 
This formula is applied to the calculation of symmetric functions, and various 
properties of the function H,,) are developed; in particular it is proved that the 
general term in /;,) is 
(—)rt# k! 
A,HA,=*HM.% «6 
!a,!a,! Late ane 
y+ Ag+ Ag+ eae 
k being the degree of the term, which gives rise to the following curious theorem, 
viz.: If in the expression of 8, we multiply each term by the degree of the term, 
and then divide the whole expression by the degree of the expression, we obtain 
Hf), and conversely. 
§ 4. The following inverse operators are obtained, viz. :— 
Vihi= 2a, + 8a, leah ait a 
sat f 2 
a,—— + 
‘day 
=8S,d,—S,d, + S,d, —S,d, + oe + (—) S410, + “uOed 
oer (ine ad . da 212 <a “1s da 
Eee puiieion cate coor ee dele aa ie sp Sly 
= 8d, —Sp4id, + Sppod.— . 2. +(—)8sSpadst oes 
and it is proved that 
V_,(X) =AQIr) 5 PX. 
V_,Q%) =(14+ IQA), 
The effect of the operator on certain other functions is considered. 
§ 5. The definition of H(,) is extended, so that the function H(,2) has a definite 
meaning, and it is shown that H(,?) bears the same relation to symmetric function 
Qi) that Hj.) bears to (A), and is derivable from it in the same manner, viz. by 
means of the theorem given ante. 
A method is given for calculating Sw.j where Sw.j represents the sum of all 
the homogeneous symmetric functions of weight w and of degree 7 at most. 
§ 6. Representing the expression 
Beatport iN) BAAN Tor OD 1) Ee gee 
+(—)sHeag(A. per. ss ]s)t 0... 4(-)A.per. as. 1) 
