DR WALLACE ON A FUNCTIONAL EQUATION. 



and by adding and subtracting, 



645 





-^(y+c(y-(y-eiy]. 



2 1 " I 2a"- 



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



•^^"^^^ 2^L { ^^''^ + F W } " + { /(=^) - F (z) } "] , 



^^""■^^^L { •^^'')+ F W } " - { /W - F W } "] . 



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

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



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



n— 1 



Viz. f{'^)^^\_[f(?^')+ 'P{nx) ]K[f{nx)-¥(nx) } ^ j ; 



n— 1 



By these formulse, combined with this, 



{f(nz)Y-[¥{nx)Y=a\ 



we may find /(«»;) and V{nx) fi-om/(s;) and F(«), and the contrary. 



35. 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=fl!, 0Q=«;, 



(1) J>(^=f(z)^y=^[f^+e-^], 



(2) ¥{z) = ta = ^[e^-e'-'], 



(3) s=^[e^-e-^]. 



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

 dinate and y, since 



dy , 1 f ^ -^1 



VOL. XIV. PART II. 



6E 



