DR WALLACE ON A FUNCTIONAL EQUATION. (543 



the curve ; we may, to distinguish the constants from each other, now name c its 

 modvlus. We have seen that a curve of equilibration has some properties ab- 

 solutely identical with those of an hyperbola, and quite analogous to those of the 

 circle and ellipse ; such is that given in Art. 27, and I shall now investigate others. 

 32. It has been found that the relation between s, the curvilineal space 

 BPQC, and t, the tangent of the angle PKQ, is expressed by the formula 



— = tc . 

 c 



Let either of these equal quantities be designated by the symbol F (x), which in- 

 dicates a function of the variable amplitude a^ ; let <r^ and x^ be any values of .r, 

 then 



will denote the same functions of 



that /(a?) and F (<2?) are of x. 



1 



To abridge, let us put r to denote e~. We have found that 



2/(^0) = «{ r^'°+ r-""") ' 



2/(^,) = «{/v+ r-^, ), 

 2F(.r,) = a{^^-c_^-'^o) , 

 2F(^,) =a{r''-r-'-'' } • 



From these formulae, by multiplying corresponding sides of the equations, 

 we get 



4/(0./(-2-v) = «'{y^^+r-^=} {r^'+r-^'} (1) 



4 F (^ J . F (a,' ) = a' { /^c _ r-''. | j ^^. _ ^-^V } (2) 



4/(.r,).F(^,) = «2{/-»+^-^o) {^^,_^-'-v| (.3) 



4F(a;,)./(.r,)=:a2{^^._^-^o} {^*.+ ^-^.} (4) 



Now, «2 { r^o + y-^o ) { y*V + y-*, ) = «' { r'^° + "'' + ^"^"'^ + ■^.) ) + a2 { r* -^V + ^-(^- -•'■.) ) : 



Again, «' { /*'= + ""' + r-(^o + *v) ) = 2 «/(*•, + a;,) , 



And «^ { /-^o-^' 4- r-(^o-«^.) } = 2 a/(^, - ^,) ; 



Therefore, 4/(^J ./(^,) = 2 a { /(^„ + ^J +/(-r,-.r,) } . 



The remaining equations (2), (3), (4), may be treated exactly in the same 

 way as equation (1), and like results will be obtained ; and from the four we ob- 

 tain these formulae 



