442 Mr. Herapath on Functional Equations. 
three rectangular coordinates of a sphere, that is, of the quan- 
tities, 
p ¥1l—pwlcosa, YW 1 — p*! sina; 
then, r being the radius of a stratum, we shall have, 
r=afiteg + ee + &o(6) + &e.}, 
e being a quantity according to which the steps of approxima- 
tion are to be arranged. This general expression supposes 
that the central nucleus is an ellipsoid ; but if it be a spheroid 
of revolution, then we must make ¢°*) equal to uw’, or 1 — p’*; 
and ¢‘#), ¢(°) &¢. must be trinomial, quadrinomial, &c. func- 
tions of pz’. The quantity e, the density of the stratum, and 
the coefficients of all the angular quantities are functions of a; 
and they must be determined so that a stratum shall satisfy~ 
the conditions of equilibrium, and, when it comes to the centre, 
shall rest upon the solid nucleus. In this manner we shall 
have the most general solution of the equilibrium of a hetero- 
geneous mass of fluid, pushed to any proposed degree of ap- 
proximation. I am, &c. 
June 14, 1826. J. Ivory. 
—— 
LXIV. Supplement to Mr. Herararnu’s Paper in the Philoso- 
phical Magazine for August 1825, on Functional Equations. 
By Joun Herapatn, Esq. 
To the Editor of the Philosophical Magazine and Journal. 
Sir, 
FROM the period of sending my last communication to the 
Phil. Mag., my mind has been so much estranged from 
the consideration of functions, that I might almost be said to 
have entirely abandoned them. ‘This is the reason why so 
long a time has elapsed without my having noticed the omis- 
sion of a restriction, which, on reperusing the paper alluded 
to a few days since, I was surprised to find I have no where 
distinctly mentioned. It should have been stated, that the 
arguments of the paper relate to those periodical functional 
equations only, whose solutions contain arbitrary functions. 
We may indeed understand this from “ arbitrary functions,” 
“‘ complete solutions,” &c. in the heading of the paper; but as 
there are an infinite number of periodical functional equations, 
having their solutions complete, without containing arbitrary 
functions, and therefore wanting the properties which render 
our reasoning applicable, I hasten to supply the omission. 
As 
