Eoss — Verb-Functions. 



71 



In this, 5 is easily the predominant coefficient ; and the approximation 

 is fairly rapid, two roots being obtained simultaneously from the form 



y- + y'^ = 5. 

 Taking only five terms of the invert, we have 



^=1+^= 3-1284 ... and =-l'33005... 



Again, putting y = ~ » "h^cvQ 



3^ - 02^ + 1 = 0, or 2 + 2"- = 5, 

 from five terms of the invert of which we obtain 



^ = 1 + i = 1-201642 . . . , 



the three roots agreeing with those obtained by Horner's process. 

 Similar artifices may be employed for many other equations. But the 

 matter is beyond the scope of this paper; and enough has perhaps 

 been written to suggest the general validity of the method. 



(8) In conclusion of this part of the subject, it is important to 

 note that in the above examples the subject of the invert itself, 

 that is, the weight-function of the predominant coefficient, gives a 

 rough indication of the position of the roots. Thus, in the last 

 example, these weight-functions alone, without the following series, 

 would give 



X = 3-0, X = 1-29 ... and x = - 1-29 . . . ; 



which are not very inexact localisations. The development of this 

 and many other points in this connexion must be left to another 

 paper. 



In the Table at the end of this paper the values of a number 

 of terms of ["^2]"^ • . • are given in order to save tlie 



trouble of calculating the coefficients. 



24. Linear Differential Equations treated hy Operative Division. — 

 A brief note on this subject must suffice. If 



[D'+p.D+p.D' . . . > = [<^> 



be a linear differential equation with constant coefficients, 



