644 DR WALLACE ON A FUNCTIONAL EQUATION. 



f{x, + ,r,) +f{x^-x) = ^f{x:) .fix:) ■ (1) 



f{x^ + x)-f{x^-x)=^F{x:}.¥{xy, (2) 



F {X, + ^,) - F {x^ - x;) = |-/(*J . F (^,) ; (3) 



F (*, + ^,) + F {x,-x) = ^ F (^„) ./(^,) . (4) 



From these again, by addition and subtraction, we find 



«-/(^o + ^,) =/W ./W + F(^„) . F(x,), (5) 



a./(a-„-^,) =/(^J ./(a:,)-F (x,) . F (a;,) , (6) 



a . F (a;„ + ^,) = F (^J ./(^,) +/ W • F (^,) , (7) 



a . F (^„ - a.,) = F (^J ./(x,) -/(^„) . F (^,) , (8) 



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

 hyperl)ola in my Memoir, already quoted, on the Analogy between the co-ordinates 

 of these CmT^es, 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 /' {x), then (Art. 30), 



KQ_/(^)_/(^) . 

 c c F (ic) ' 



onH / (-^o + ^0 _ /(^, + ■^.) _ /(^o) f{^) + F (.r.) F {x) 



F(^c + ^,) F(a:,)/(^0+/(^„)F(^,) • 



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



Therefore, substituting and dividing the numerator and denominator by 

 F (.rj F {x,), we find 



/(.. + .,)_ ^,^^^^^^,^^^ . (9) 



And simUarly, /' {x^-x} = ^'}"{^^^!:f~^ • (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. 



i V + ct -- y — ct 



e' = - , e '=^ ; 



a a 



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



