the tension of overhead wires supported by equidistant poles. 207 



which is accordingly a superior limit to the error involved if we 

 neglect the powers of x above the mth in the expansion of y, 

 provided 



We have in fact for such values of x 



T [1 — -rp — ai^f^„ — as 



\/l — 2x <y <{l — x). 



If now we take a value of x less than ^ and form the series 



T (1 — X — a^x^ — a^x^ — • • •) 



it will represent some value of the function T^ where, however, 

 n is unknown and is not necessarily an integer. 



We can similarly form the series 



\Aj %Aj *Aj 



which will also converge and will represent T,i+i. 



Now Tn is positive and the series which represents it is 

 absolutely convergent and therefore its square forms a convergent 

 series with the same radius of convergency. Now it is known (see 

 Bromwich's Theory of Infinite Series, p. 216) that the circle of con- 

 vergence of the reciprocal of a power-series is either the same as 

 that of the original series, or else reaches up to the zero of the 



given series which is nearest to the origin. Thus jfj-^ forms a 



convergent series, since the absolute value of x does not extend as 

 far as a zero of the function T^. 



We thus see that 



a + 2.0 In — jf-^ — In+i 



forms a convergent series in x. 



But this will represent Tn-i which is accordingly given by our 

 series and must be absolutely convergent so far at least. 



If now it should so happen that Tn-i is positive, we may repeat 



this process and ™ — - will give a convergent series in x and 



-^ n—i 



therefore T„_2 is given by a convergent series. 



If, however, Tn-i is negative we have passed over a root of the 

 equation 



1 



and consequently ™ — - will no longer converge and the series 



will not give Tn-^. 



