058 DR 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 =/ (^o)/(^-) + F W F W ; F (^„ + x) =F (^ J/ W +/(^c) F (^,) ■ 



Thus the first and second terms of a series of values of f{x) and F (cc) were 

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

 gated as follows. 



In the fovm\Aa,/{x, + A)+f(x^-h)=2/{x;)/(h), put x + h instead of a?^, and we 



have 



f(x + 2h)+f{x)=2fix + h)fQi) : 



Now, /W = 1 + T:2+ 1.2.3.4 + 1.2.3.4.5.6 + '^''■ 



Let x=x + h, z^=x, + h, x^=x., + h, &c. be successive values oi x, which go on in- 

 creasing by differences, each equal to h, and put 



P='' + XT+ 374^ + sTlv/eTTs + ^"^ 

 Then, from what has been just shewn, we have 



/(:r,)+/(r) = 2/(:r,) + P/W, 



and /(=^.)=/W + {/W-/Wl + P/W; ) 



similarly, / W =/W + {/W -/(«,) } + P/W, )■ («) 



and /W=/W + {/(^3)-/W} + P/W> ' 



Thus, all the numbers in the series f (x), f {w';), f {x^, &c., which follow the first 

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

 P/(a;,), P/(«2), P/G^a)) ^ave been found. In the computation of the tables, h was 

 assumed to be 1, or jSj or joo, or ijoso- 



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



When A=l, then Vt=t+ ^-^ + g ^ g g + &c. 



1 _ t t < „ 



When A= jQ, P '- JQQ-+ gQQ 4QQ + 3QQ 400 . 500 . 600 ^'^ 



&c. 

 These series converge very fast, and their terms are readily 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 K + A) + F ix^-h)=2f{h) F («J, which, putting x + h for x,, gives 



F (a! + 2A) + F (a!)=2 F (« + /i)/(A) : 



Hence, putting x=x + h,x^=z,+h, he, and P for the same series as before. 



we have 



F (x.^ = F (^,) +{ F (;»:,)- F a;} + P F (:r,) (fi). 



