TRANSACTIONS OF SECTION A. 263 



expression which, when multiplied by y, is equivalent to x. The nianifoldness 



x 

 of the arithmetical value of - follows from the circumstance that in zy - x the y is 



co-ordinate with, not subordinate to the s. 



The expression x m denotes the selective operation resulting from m of the x 



operations being applied together ; and similarly x*r> denotes that selective opera- 

 tion which is such that when m of it are taken simultaneously the result is 

 identical with x. 



The rule of signs follows from the relation connecting + and — , viz., 

 + 1 — 1=0; taken together with the restriction of + to denote no change of 

 direction by defining + 2 = + . 



Since an expression is in general both positive and negative, an equation in 

 general involves two component equations, the one of which refers to the positive 

 part and the other to the negative part. Hence an inequation requires in general 

 two signs. Thus a — by £x—y asserts that the positive part of a — b includes the 

 positive part of x — y, and that the negative part of a — b is included in the negative 

 part of x — y. The ordinary axioms concerning the transformation of equations 

 and inequations still hold true. 



It follows from these principles that there is an Algebra of Quality which 

 absorbs the ordinary theories of necessity and probability, and that this Algebra is 

 a generalised form of the ordinary Algebra. Hence all the theorems about 

 quantity are, after being properly generalised, true of quality also ; and conversely, 

 all the novel theorems about quality are, after beiDg restricted by a particular 

 condition, true of quantity. 



5. Note on a Theorem in Linear Differential 'Equations. 

 By W. H. L. Russell, F.R.S. 



The author after calling attention to the circumstance that a linear differential 

 equation of the second order is immediately integrable, if the coefficient of the last 

 term taken negatively is equal to the sum of the two first terms, gave the following 

 theorem : — 



Let H^ + K €* + L £!? + M — + N = 0, be a linear differential of 

 dx* dx 3 dx 2 dx 



the fourth order, where H, K &c. are rational functions of x, then if z = 



y-72 si 



\ -=-^ +jj^ + v w, the proposed equation may be reduced to linear differential 



(IX' (t iV 



equation of the second order in s, if 



N V - 2LW + (L 2 N 2 + KMNV + (2HLN 2 - KLMN - K 2 N 2 - HM 2 N)p 5 

 - (2H 2 N 2 + 2HL 2 N - 2HKMN - K°-LN - HM 2 L)p 4 

 + (2H 2 LN - HKLM - WW - HK 2 N)p 3 + (H 2 L + H 2 KM)p 2 - 2H 3 L P + H* = 0, 

 where p is any constant. 



6. On the Repulsion of Wires influenced by Electric Currents. 

 By W. H. L. Russell, F.B.S. 



The object of this paper was to ascertain the possibility of a certain experiment 

 for ascertaining the repulsion of two voltaic wires influenced by currents moving 

 in them in opposite directions. 



7. On Plane Class-Cubics with three Single Foci. 

 By Henry M. Jeffery, M.A. 



1. These cubics may be studied in three divisions, as the triangle ABC formed by 

 the foci as angular points is equilateral, isosceles, or scalene. The cases have been 



