162 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



5th convergents of the. continued fraction f b \ Since we find that to the degree of 

 accuracy retained these are equal, it follows that all subsequent convergents are 

 sensibly equal to either of them. Hence the infinite continued fraction /j 1 may be 

 replaced by its fifth convergent without sensible error. 

 Similarly we find 



log F} = 0-1834, log E.l = 0-5045, logFJ = 0'8266, 

 and therefore 



Mi FJ /j 1 = 9753 - 6707 - 3"636 = - 0'590. 



As a second trial we take 



\/<o - 2-4600. 



Proceeding as before, we deduce 



Mi - FJ -/ 5 ' = 10-234 - 6-619 - 3'607 = O'OOS. 



We conclude that there is a root of the period-equation lying between 2*4400 and 

 2*4600 ; by interpolation its value is found to be 



2-4597. 



The same method may be used for the determination ot the roots of the second 

 class, the initial trial values being suggested by the formula (63). As a numerical 

 example, if we put n = 5, s = 1, hg!ara- = -^ in (63), we find 



2&>/X = 40-974, or X/co = 0'04881. 



For a first trial we take 2w/X = 41, and deduce 



N EJ - ej = 11 + 8*678 + 2'602 = 0'280. 



As a second trial we take 2w/X = 41*280, and obtain 



N! - Ei - el = - 11-280 + 8*657 -f2'593 = - O'OSO, 



and therefore, by interpolation, 



NJ - E> - el = 0, 

 when 



2o>/X = 41-253, or X/co = 0*04848. 



I have selected for special investigation the asymmetrical types when s = 1, and 



