494 Royal Irish Academy. 
In fact, the distinction between the plane of incidence and the per- 
pendicular plane no longer exists, and the phenomena must be the 
same in all planes passing through the ray. Yet M. Cauchy, in the 
two places above quoted, asserts it to be a consequence of his theory, 
that in this case the alterations of phase are different for two planes 
of polarization at right angles to each other, and that the difference 
of the alterations amounts to half an undulation. The same singular 
hypothesis had been previously made by M. Neumann (Poggendorff’s 
Annals, vol. xxvi. p. 90), whom M. Cauchy appears to have fol- 
lowed: but M. Neumann has since admitted it to be erroneous 
(Ibid. vol. xl. p. 513). 
The Chair having been taken, pro tempore, by the Rev. J. H. Todd, 
D.D., V.P., the President (Sir W. R. Hamilton) communicated the 
following proof of the known Jaw of Composition of Forces. 
Two rectangular forces, v and y, being supposed to be equivalent 
to a single resultant force p, inclined at an angle v to the force z, it 
is required’to determine the law of the dependence of this angle on 
the ratio of the two component forces x and y. 
Denoting by p’ any other single force, intermediate between x 
and y, and inclined to x at an angle v', which we shall suppose to be 
greater than v; and denoting by 2’ and y’ the rectangular compo- 
nents of this new force p’, in the directions of 7 and y, we may, by 
easy decompositions and recompositions, obtain a new pair of rect- 
angular forces, 2!’ and y!’, which are together equivalent to p’, and - 
have for components 
ate + Ly; 
2 B 
R Ys 
yi= tye 
He 
the direction of x! coinciding with that of p', but the direction of y" 
being perpendicular thereto. Hence 
y" i x y'— y a! ; 
a eat yy 
" ' 
that is, tan—1Y_ = tan™ is Eitan FS: 
z x x 
or, finally, f('—v) =f(r)—-f(v), «2 ees (A.) 
at least for values of v, v', and v'—v, which are each greater than 0, 
and less than > ; if f be a function so chosen that the equation 
4+ = tan f (v) 
2 
expresses the sought law of connexion between the ratio a and the 
angle v. The functional equation (A.) gives 
f (mv) = mf (v) = = f (nv), 
m and n being any whole numbers; and the case of equal compo- 
