2 3 o SCIENCE PROGRESS 



vergence will be slow ; but in this case multiplicity will be 

 suspected, and may be detected by other means. 



As an example of how transference of origin will change a 

 divergent into a convergent invert take the equation mentioned 

 at the end of last sub-section, namely o = i — x — x % . The 

 invert of this is x = I — I + 2 — 5 +14 — 42 + etc. Now 



as the root evidently lies between -- and 1 , transfer the origin 



mt 



to the former point and we have the equation o = 20 — z l ; 



4 



which gives, if z — -y, = 1 — y -y*. From this 



o IO 



y = ■ - 75 + 2 (h) ' ~ 5 (h)' + etc - = ° • 944- • • • . 



and z = o • 1 1 80 . . . , and #=o*6i8o...=-( V5 ~" l ) • 



These preliminaries being settled, let the general equation 

 be written 



[0 - co 2 + do 3 - eo l + /o* - go 3 + . . .] x = a ; 



so that x — [0 — co 2 + do 3 — eo* + /o s — go 8 + . . .] -1 <z. 

 Proceeding to divide, 



o-co' + do 3 -. . .]o [o + co 2 + (2c 2 -d)o J + (sc s -$cd + e)o i +etc. 



— co 2 + do 3 — . . . 



co 2 — do 3 + 50* — . . . 



CO 2 - 2C S 5 + C (C 2 + 2^)0 4 — ■ . . 



(2c 2 - rf)o J - (c 5 + 2cd - e)o l + . 

 (2c 2 - d)o 3 - (2c 2 - d) 3co 4 + . 



(5c 2 - $cd + e)o K + . . . 



It is advisable to write out the successive powers of the 

 divisor before beginning the division. We have finally, 



x = a + ca 2 + (2c 2 - fiOa 5 + {5 C ( c ' - «0 + ^}« 4 

 + {14c 4 - 2ic 2 d + face + d l ) — f}a l 

 + {42c 6 - %4c 3 d + 2Sc(ce + d*) - 7(cf + de) + S r )a , + etc. 



This series is given with weighted coefficients and directions 

 as to how to obtain any more terms (which we may already 

 suspect to be invariant) in "Verb Functions," pp. 52 and S3 

 especially. Together with the particular instances of it given 

 in sub-section (2) above, it is already known, being obtained by 



