648 DR WALLACE ON A FUNCTIONAL EQUATION. 



of the equilibrated curve, is equal to a, its parameter ; and, in addition, we shall 

 generalise the properties given in this article. 



\^ 

 40. Putting 7'=e% {e the base of Neper's logarithms), the equations of the 



catenary are, 



f{x) being the ordinate, and F {x) the arc, corresponding to the amplitude x. 

 Let x^^ x^, x^, ... x„, be any values of x, we have 



ar^-^ =/(x,) + F(a;,), ar"*'^ =/ (^2) -F (;.,), 

 ar^3 =f^^^^Y{x,), ar-^^ = f{x,)-Y (x,), 



• • • • 



The sum and difference of the products of the sides of these two sets of 

 equations being taken, and it being observed that 



ar—^''^'*^''>-^'''^=f{x, + x, + x,hc.)±V(x, + x, + x,kc) 

 we have, by substituting, 



{/{x) + F (x) } {fix,) + F {x,) } {fix,) + F ix,) } &c. 

 {fix) - F ix) } {fix,) - F (z,) } {/(;r3) - F ix,) ] &c. 



{fix) + F (a:,) } {fix,) + F (;.,) 1 {fix,) + F (;C3) 1 &c. 



*^^ U{/(^,)-F(^,)}{/(^.)-F(x.)}{/(;r3)-F(af3)}&c. / ^'^ 



2a . F(ar, + a;2 + 3;3 + &c.) = s > (o) 



^ ' ^1 -{/(^.)-F (:«.)} {fix-:)~Yix,)} {/(^3)-F(;.3)} &c. / ^ ^ 



These two formulae comprehend in them, as particular cases, the expansions 

 of /(*'o — "^z) ^^d F {x^r^x,) given in art. 39. 



41. Let ^o 02, ^3, . . 0„, denote the angles which lines touching the curve at 



the tops of the ordinates f{Xj), f{x,), /{x,), . . .f{x„), make with the horizontal 



axis ; and let (p denote the angle which the tangent at the top of the ordinate 



f{x, + a?2 + «3 . . . +x„) makes with that axis ; 



ii ' 



because « • ^" =/(»;/) + F («,) = a (sec (p, + tan (p) = a tan (45° + 1 cp) 



therefore x,=a log tan (45° + i(p), 



X2=a log tan (45° + ^ ^2 ) ? 



x,=a log tan (45° + ^03)) 



• • « • 



x„=a log tan (45° + ^(p„). 



By adding into one sum the sides of these equations, and observing that 



x, + Xi + X3...x„ = a\og tan (45° + ^(p); 



