g58 ^^ WALLACE ON A FUNCTIONAL EQUATION. 



54 The values of/(l.l), F(l.l); /(.ll), F (.11), &c. were found from the 

 formulae 



/(^o + ^-.)=/(0/W + F W F (:r,) ; r(;r^ + ;.0=F(O/W+/W F W- 



Thus the first and second terms of a series of values of /(j:') and F (a?) were 

 obtained ; from these the following terms were deduced, by a formula investi- 

 gated as follows. 



In the formula/(a;^ + A)+/(a;^-A)=2/(a:^)/(A), put x + h instead of <r^, and we 

 have 



Now, /W-l + T3^ 1.2.3.4 + 1.2.3.4.5.6 ^^" 



Let x=x + h, Xi=x, + h, x3=Xi + h, Sec. be successive values of <r, which go on in- 

 creasing by differences, each equal to k, and put 



Then, from what has been just shewn, we have 



and /(«0=/W + {/(^.)-/(^)} + P/W; ) 



simUarly, / (^s) =/(«.) + {/W -/(«,) } + P/ (a:.), > («) 



and /(^.) =f{x,) + {fix,) -f{x,)} + Vf{x,) , J 



Thus, aU the numbers in the series f (x), f {x), f {x^, &c., which follow the first 

 two, are derived from them simply by subtraction and addition, after the terms 

 P/(a;,), ^f{x^, P/(«3), have been found. In the computation of the tables, h was 

 assumed to be 1, or i^, or 4> or iusoo- 



Let ^denote any term in the series of values f{x),f{x),f{x^, &c. 



When A=l, then P^=;+-g— ^+ ^ + &c. 



1 p _ ^ t t 



When A=j^, ^~^m'^m^7m'^'^^~mT^mT^'' 



&c. 

 These series converge very fast, and their terms are readOy found each from 



that before it : thus, is found from —^ by dividing the latter by 1200, 



and so on. 



55. For the corresponding series of arcs of the catenary, we have this for- 

 mula, F (iTo + /O + F {x^-h)=2f{h) F {xX which, putting x + h for x^, gives 



¥ {x-v2h)-v'F {x)=2Y {x + h)f(Ji) : 



Hence, putting x,=x + h,Xi=z,+h, &c., and P for the same series as before, 



we have 



F(x,) = F(x,)+{F{x,)-Fx] + PF{x:) (^). 



