DR WALLACE ON A FUNCTIONAL EQUATION. 



659 



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

 F (Xi), F (Xi), &c,, are to be found from ¥ {x) and F {x^), exactly as /(x^), /{xs), 

 &c. are fromy(a7) and/{x,). 



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



The calculation may stand thus : 



/(O) 1.000000000 



F(0) 0.000000000 



a=F(.l) 0.100166750 



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



b= -^ 1001668 



100 



6 



1200 



c 

 3000 



835 

 



A=/(.3) 1.045338516 

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

 B=-4r ' 10453385 



C = 



A 

 ■ 100 



B 

 '1200 



3000 



8711 

 3 



/(.4) 1.081072374 



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 figures correct throughout the whole ; but the principle of calcula- 

 tion was the same as has been here explained. 



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

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



