231 
or, since dk,, dk,, dh3, are very small quantities, 
Uz = Cy (hy” + why dk,) + CO, (hey + 2h*" dky) + Cs (Keg + wks! dks), 
or, 
Ux — (Ch? + (Of + C3k3") + x (O47 Ok, cf Ck! ok, + Cb dk). 
There is no difficulty in determining the corresponding corrections to be 
made on the values of v, and w,. 
As regards the method exhibited in this section, it may be allowed 
to state that it is precisely similar to that employed for the solution of 
systems of simultaneous differential equations, and, as I believe, for the 
first time published in a treatise on the ‘‘ Calculus of Operations,”’ in the 
year 1855. 
Posrscriet.—Upon communicating some of the results contained in 
this section to the Rev. Dr. Lloyd, with a view to the suggestion by him 
of some physical applications, if such existed, I received from him 
a statement of his views, which it is right to lay before the Aca- 
demy. He points out how, in the present position of physics, the 
conditions of the problems discussed are, in general, first treated with 
respect to some one predominant element, and then the other elements 
taken up and dealt with as residual phenomena, or disturbing causes. 
Dr. Lloyd states his belief that any attempt to apply a more rigorous 
method (in which all the elements are simultaneously taken into ac- 
count) would fail, in consequence of the large errors, which the errors 
of the observed results would entail, in the process of elimination. 
Professor Haughton has expressed his coincidence in this view. Asa 
matter of course, I defer to the opinions expressed by physicists so 
justly distinguished as these gentlemen, and put forward the methods 
contained in this paper simply for their mathematical value, whatever 
that may be. One remark only I would venture to offer. Is it not pos- 
sible that we may yet be enabled to state, by such systems of equations, 
the conditions of phenomena which depend on the simultaneous action 
of heterogeneous laws: for instance, those of heat, electric action, and 
chemical affinity, supposing for a moment the division of these agencies 
to be logically just? The geometrical illustration proposed in the first 
article would possibly give some reason to hope that this calculus of 
finite differences may yet be made more ancillary to physical research 
than it hitherto has been. 
_ The Szcreranry read the following letter from the Right Hon. Ed- 
ward Cardwell, addressed to the President :— 
*« Dublin Castle, 22nd October, 1859. 
“‘Srr,— In reference to your letter of the 17th ultimo, I am directed 
_by the Lord Lieutenant to acquaint you, that the Lords Commissioners 
of Her Majesty’s Treasury have been pleased to sanction a grant of two 
hundred pounds in aid of the completion of a Catalogue of the Museum 
of the Royal Irish Academy. 
