240 



Sir William Thomson on an Alleged Error 



has not warned us of this ; on the contrary, his instructions, lite- 

 rally followed, would lead us simply to calculate K 4 by his con- 

 tinued fraction and then to calculate K 6 , K 8 , &c. successively 

 from K 2 and K 4 by successive applications of the formula (6) 

 of § 5. This, except with infinitely rigorous arithmetic, will 

 bring out for very large values of k not the true rapidly dimi- 

 nishing values of the coefficients K 2jfc , TL 2 k+2> & c -.> but slug- 



gishly converging values corresponding to the ratio R& = —; — -, 



But this dissipation of accuracy is avoided, and at the same time 

 the labour of the process is much diminished, by using for the 

 ratios the values already found for them in the successive steps 

 in the calculation of the continued fraction for R 1# 



19. The law of variation of K v R 2 , &c, considered as func- 

 tions of e (§ 8), is of fundamental importance. Some of the 

 remarkable characteristics which it presents have been already 



mately the greater root of the quadratic, or approximately equal to — . 

 Thus, for example, take the equation 



of which the roots are 



^=•171573 and #=5'828427. 

 To find successive approximations to the smaller root, take 



r = 0, 



6 — r„ 



r 2= a-T 

 o— r. 



6-r 2 



1 

 6 — r 3 



1 

 — r„ 



r 6 = ,- 



b — r, 



•1667, 



•1714, 



= •1716, 



= '1716, 



= '1716, 



= •1716, 



r 7 =-^ ='1716, 

 b-r n 



r s = 77 



6-r 7 



1716, 



= 5-8284, 



r\=6-4 =5-8284, 

 r' =6- -L =5-8277, 



6-1=5-803, 



r', = 6- 



:5-067, 



6_-L=l-072, 



•2029, 



r' =6--l ='1725, 



r' 



r'='1716. 



If the arithmetic at each step had been rigorous, we should have found 

 r' 1 = r 1 , r' 6 = r e , and so on ! Instead of coming back on the value assumed 

 for r , we find r' = 5 8284, the greater root of the equation! 



