80 
i dy* 
§s (1+q)— pat} ga + $reg) +p 2) =0. 
(See Lacroix’s Differential and Integral Calculus, 4to. Tom. I. p. 
576, 2d edit.) ‘The result of this substitution will, for central 
surfaces, be 
dy? y A—a’ a(A°—s)F \ dy a—a’ 
= (2 4 0 
dx? ev A—A yo AA (A—A) ay SF dr Aa—aA 
having eliminated 2° by means of the equation of the surface. The 
substitution, for the surfaces which have no centre, gives 
dy? ( a ce? c? a’ — c? ('—a)\, === 0 
v iat yo 775 aay —_ —— 
These two equations are of the same form, and may therefore be 
integrated by the same methods. Let them be represented by the 
general form 
for surfaces having a 
in wick 
A=— 
A’ ce” (A"—a’) 
centre, and e= yand i rr for surfaces having no centre. 
This might be integrated by resolving it into its factors, but 
more elegantly by ascending to a differential equation of an higher 
order. For this purpose let this equation be differentiated. The 
result is 
