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 ?'=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 a;, we have 



ar''- =/(^,) + F(:t,), a^""^' =/(;>;,)-FW, 

 «r^. =/(;r.) + F (;.,), ar"*^ =/(^,)-F(a;,), 

 a/-^3 =/(a;3) + F(;E3), «»--^» =/(;r3)-F(a;,), 



ar*»=/(;r„) + FW, a/-*" =/W-F(;t„), 



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

 equations being taken, and it being observed that 



we have, by substituting, 



""""'■^■^^'^^^^^'"-^"^^ U{/W-F(.,)}{/(..)-FW}{/(-3)-F(.,)}&c. ) ^^^ 



""""•^^^'^^^^^'^'^''•^^ l-(/W-F(.,)}{/(..)-F(.,)}{/(.3)-F(.3)}&c. ) ^'^ 



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

 Qif{x^±x^ and F {x^^x^ given in art. 39. 



41. Let <p,,<p2,(p3, ■■■((>„, denote the angles which lines touching the curve at 

 the tops of the ordinates f{x,), /(x,), f(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, + x, + x,... +x„) makes with that axis ; 



because «.«"" = /(«,) + F(a;,) = o (sec ^, + tan </),)=« tan (45° + ^ i^,) 



therefore x, = a log tan (45° +i(p^, 



a;j = a log tan (45° + 1 ^2 ), 



a-j = a log tan (45° + i ^3) , 



x„ = a log tan (45° + ^ ^„) . 

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

 a:, + a;a + »:3 . . • «„ = alog tan(45'' + i^) ; 



