DR WALLACE ON A FUNCTIONAL EQUATION. 645 



and by adding and subtracting, 



■4 



These formulae, by our functional notation, may be expressed thus : 



/(««^)=2^[{/(^)+FW)"+{/(^)-F(«:)}"], 



They denote a property of the curve of equilibration quite analogous to that 

 of the conic sections which is expressed by Demoivres theorem. From these 



formulae, by putting n x instead of x^ and — instead of n, we obtain two others, 



n— 1 



viz. f{^)^^\_[f{nx)-^Y{nx) \^ ^ [f{nx)-Y {nx) ) ^ j ; 



n— 1 



By these formulae, combined with this, 



[f{nx)Y-[Y{nx)Y=a\ 



we may find/(na;) and Y{nx) from/(;c) and F(a;), and the contrary. 



85. As in an ellipse or hyperbola, which, like a curve of equilibration, have 

 two parameters, if these be supposed equal, the curve becomes a circle or equila- 

 teral hyperbola, which have each only one parameter : so, in like manner, we 

 may assume that a and c, the parameter and modulus of a curve of equilibration, 

 are equal. Then the equations of the curve are a little more simple, they being, 

 putting BC=:a, OQ=a;, 



(1) PQ=/(^)=^=|-{«" + e-^} , 



(2) Y{x) = ta = ^[i^-e--^\, 



(3) *=|-{e^-e-t| , 



/=(l + 0«^ . g ^ 



In this case, putting z for BP, the length of the curve between the least or- 

 dinate and ?/, since 



dy ^ 1 . £ _£, 



VOL. XIV. PART II. 6 E 



