692 
MR. A. McAULAY ON THE MATHEMATICAL 
10. The following easily-proved propositions should be noticed :— 
Prop. V. If Cl be of Class I., then H -1 is of Class II, and if T be of Class II., 
then T _1 is of Class I. 
Prop. VI. Her is a flux, and Tr is an intensity. 
Prop. VII. SV (Do-) ds = SV' (flV) cW. (Props. IV. and VI.) 
Prop. VIII. Cl'dS' is the same function of CldS as dp' is of dp, and T 'dp is the same 
function ofTdp as dS is of d% or Cl'dS = yflcZS, T ip' = mf~ l Tdp. 
Prop. IX. Cl'A'dI = yflAcG. [Prop. VIII. and eq. j(4).] 
11. Going, now, back to our definition of electric coordinates (§ 4), since for each 
element they may now be written SD d$ a , SD d% 6 , SD dt c , and since dt a , &c., are constants, 
we see that the choice of coordinates is equivalent to regarding D and not D' as the 
independent electric variable at any point. Further from eq. (1) § 4, and eq. (6) § 7, 
we have 
SC d% = dSDdS/dt, 
or, since d S is an arbitrary constant vector, 
C = dB/dt .(11), 
which is, of course, inconsistent with the equation C' = clD'/dt. 
C.—Preliminary Justification of the Foundations of the Present Theory. 
12. I have deliberately led up as quickly as possible to a description of the mathe¬ 
matical machinery to be used subsequently, as it has been necessary to notice 
incidentally some of the essential chai'acteristics of the fundamental assumptions and 
the methods of investigating their consequences advocated in the present paper. As 
a preliminary justification of these assumptions, I cannot do better than indicate the 
line of thought which led up to them. 
In studying Maxwell’s theory, and seeing how beautifully it was built up step by 
step from a mass of experimental facts, till the consistent whole stood revealed, it 
seemed to me that, notwithstanding the general harmony of its different parts, there 
was just here still something to be desired, some single plan that should govern the 
whole. This statement may not seem justifiable, so I instance two examples of the 
want of harmony. In one part of his treatise, the kinetic part, he works out the con¬ 
nections between the different parts of his theory by the general methods of 
Dynamics. But not so in the statical part. It would seem that the statical part of 
the subject, in such a plan as just mentioned, ought to appear as a particular case of 
the kinetic, whereas, in Maxwell’s treatise, the statical terms in the equations are 
merely added on to those deduced from dynamical reasoning. The same remark 
applies to the terms necessary to produce the mechanical effects of magnetism 
