. TRANSACTIONS OF SECTION A. 411 
by F{@®.A.p.r .. .)}, the function Fis examined, it being first proved that it 
is in every case a non-unitary symmetric function ; numerous relations are deduced, 
and in particular an expression is obtained for the sum of all the non-unitary 
symmetric functions of a given weight ; this is found to be 
=2 
3(—)* How» (2 od?) (2? 784) + (28 ee). 2 4 (2%) or ((216=D1)}> 
s=w 
Also the sum of the non-unitary symmetric functions of weight w, which contains 
no part less than 7, is 
3 Hoi) + (-YAGVD} — How-g-n (GD) + (-YGV)} 
+ Hee-0-9{(1) + (=) — 
where & is the integral part of “, and the terms are continued up to and including 
the term containing H,. 
§ 7. The expressions in terms of the coefficients of the non-unitary symmetric 
functions, or, as they may be otherwise called, the single partition semimvariants, of 
weight thirteen, are given in a table. 
9. On the most Commodious and Comprehensive Calculus. 
By Dr. Ernst ScHRODER. 
The calculus of the four algebraical operations—viz., addition, multiplication, 
and converse—does not imply the greatest abundance of formal laws and conse- 
quences. Accepting symbolically the notation a+ for a certain function f (a, b) 
of two variables, and a. or ab for another function ¢ (a, 6), a calculus is found 
out to be the most comprehensive one that enjoys the following properties. 
Both functions are throughout one-valued (=determinative) and invertible ; 
they are commutative and associative, like the proper addition and multiplication ; 
besides, they submit to the law, that whenever a+6=c, then also 6+c=a and 
e+a=h; again, when a.b=c, then also 6.c=a, and c.a=6, so that every term 
or factor may also be transferred as such to the other side (member) of any 
equation. 
None of the distributive principles, as a .(6+c)=a.b+a.c, holds good between 
the aforesaid operations, but instead of these we have the still simpler and ampler 
law expressed by the formule 
(a+b).c=(b+c).a=(c+a).b=c+ab=b+ca=a+be, 
Brackets in this calculus are therefore obliterated by simply cancelling them, 
which yields the most commodious imaginable way of developing products of poly- 
nomials. Moreover, in a series of operations it is allowed to confound any two 
signs of addition and of multiplication. 
Lastly (the formule 
a+O=a, a 
e007 
proving generally valid), the solution of an equation is most easily performed, when- 
ever possible. 
It is not difficult to explain the functions f and ¢ for the whole dominion of 
ordinary complex numbers, both proving generally discontinuous. First explain 
them for a province of but two numbers, say 0 and 1, by means of the tables 
20] O= 144. rO=0.1=156 
1=14020+1 eb sonid 
