DR WALLACE ON A FUNCTIONAL EQUATION. 



659 



This formula, compared with formulae («) in last article, shews that the arcs 

 F {X2), F (^3), &c , are to be found from F (x) and F (.3?,), exactly as /{ao^), /{x^), 

 &c. are fYom.f{x) andy(a7^). 



56. As an example, let it be required to find the numeral values of the series 

 of ordinates/(.2),/(.3),/(.4), &c., and arcs F (.2), F(.3), F (.4), &c. having given 

 /(0) = 1, /(.I) = 1.005004168; F(0)=0, F(.l) = 100166750 



The calculation may stand thus : 



/(O) 1.000000000 F(0) 0.000000000 



A=/(.l) 1.005004168 

 /(.l)_/(0) 5004168 



B= A 10050042 

 100 



8375 

 3 



C= 



100 



B 



1200 



C 

 3000 



a==F(.l) 0.100166750 



F(.1)-F(0) .100166750 



— 1001668 



100 



iioo" ^^^ 



3000 " 



6= 



€ = 



A = /(.2) 1.020066756 

 y(.2)-/(.l) 15062588 



ffl=F(.2) 0.201336003 

 F(.2)-F(.l) .101169253 



B= 



C= 



100 



B 



1200 



C 



3000 



10200668 



8501 



6 = 



<r = 



a 



loo" 



6 



1200 



c 

 3000 



2013360 



1678 







A=/(.3) 1.045338516 



/(.3)-/(.2) 25271759 



B= 



C = 



100 



B 



1200 



3000 



10453385 



8711 



3 



a=F(.3) 0.304520294 

 F(.3;-F(.2) .103184291 



6 = 



c — 



a 



Too 



h 

 1200 



c 

 3000 



3045203 

 2538 

 1 



/(.4) I.O8IO72374 



F (.4) 410752327 



These values of/(.2), /(.3),/(.4), and F (.2), F (.3), F (.4), are true to seven 

 decimal places. In this way tables I. and II. were constructed ; but the values 

 were found to more decimal places. Precautions were also used as checks to 

 bring out ten figm-es correct throughout the whole ; but the principle of calcula* 

 tion was the same as has been here explained. 



57. The Tables which are to foUow require hardly any explanation. In them 

 all, the parameter, that is/(0), is unity. The first gives the values off{x), F {x)^ 



