INTO ARITHMETICAL EXPRESSIONS FOR INFINITE SERIES. 187 



But 0«.r, considered as the limit of c^'a;, may be one root of 

 (px = ,%; for values of x intermediate to one set of limits, and another 

 for another. For instance, let p be greater than unity, and let 

 (px = xP. Then we have 



(p"x = x''" = when a;< 1 = 1 when or = 1 = oc when x>l. 



We might generalize the theorem, by a supposition which common 

 algebra would admit. An equation of any degree, considered as one 

 of a higher degree with evanescent terms, has injimte roots. In the 

 common mode of speaking, we must say that <p^-x is either infinite, 

 or a root of cpx = x. In that just alluded to, we should simply say 

 that 0°=x is a root of (px = x. 



As a second example, let (px = a (x + tn) ~ m. 



Then 0-a; = a°=(x + m) - /«, 



and is infinite for all values of x except - m, when a is greater than 

 1, and = - 7«, for all values of .r, when a is less than 1. But - m 

 is the root of (px = x. 



It must, I suppose, be well known that successive approximation 

 will not be vitiated by any error introduced into the approximate 

 results, unless that error be so great that the process is made to tend 

 towards another solution, or to increase without limit. For instance, 

 in the solution of 



1 



* = I : ^ by a contmued fraction. 



The value of x may be what we please at the commencement, 

 or the obtained value may be altered; and the attainment of any 

 degree of accuracy, though retarded, is not rendered impossible. In 

 a similar manner, a purely graphical process will lead to information 

 upon the value of <^"x in particular cases, such as with a little care 

 may be made equivalent to demonstration. Let OA be a line equally 

 inclined to OX and OV, rectangular axes to which the curve y = <bx 

 is referred. Taking any point P,, which has the abscissa required, x, 



AAS 



