(544 DR WALLACE ON A FUNCTIONAL EQUATION. 



/(^o + ^) +/('^o -^.) = —f{x:).f{x) ; (1) 



/(^o + '^.)-/('^o-^.) = |-F(^o).F(j:,); (2) 



F(^„ + ^,)-F(a;,-^,) = -|/(^J.F(^,); (3) 



F(^, + ^,) + F(^,-^0 = 4f(0 ./(^.) • (4) 



From these again, by addition and subtraction, we find 



a ./(^, + ^,) =/(^,) ./(^O + F (^J . F ix;) , (5) 



a./(;r,-ar,) =/(^,) ./(^,)-F (^<,) . F (^r,) , (6) 



a . F (^„ + ^.) = F (:r J ./(^^ +/(a;,) . F (^,) , (7) 



a . F (07^ - a;,) = F (^,) ./ (^,) -/(^,) . F (^,) , (8) 



These formulae are absolutely identical with those given for the ellipse and 

 hyperbola in my Memoir, already quoted, on the Analogy between the co-ordinates 

 of these Curves, the variable line x here coming in the place of the elliptic or 

 hyperbolic sectors. 



33. Considering the subtangent KQ as a function of the amplitude <r, let it 

 be denoted by the symbol /' {.v), then (Art. 30), 



KQ-/(^0_/(^) . 



c c F (a;) ' 



«nH / (-^o + ^0 - /(^o + ^) _ /(^o)/(^.) + F (^J F (x) 



F(^o+^o Y(x:)f{x)+f(x:)Y{x) • 



Now, / W = ^^^^ , and /(..) ^ ^-^^ : 



Therefore, substituting and dividing the numerator and denominator by 

 F (*J F {x^), we find 



And similarly, /' {x^-<v) = ^' ^!';}( ^1^7"' ' (10) 



All these formulae are perfectly analogous to properties of the ellipse and 

 hyperbola, particularly the latter of these two curves. 



34. Resuming the formula of Art. 28, viz. 



- y + ct -- y — ct 



e'=- , e •'=^ ; 



a a 



let n be any number whatever, positive or negative, whole or fractional, then 



{^} 





