366 Mr. J. Prescott on the 



small fraction . It is clear, then, that the series first converges, 

 then diverges, and finally converges absolutely. The series 

 will be useful if we can show that, by stopping at the point 

 where it first converges, we get a good value for <£. This 

 we shnll now prove. 

 64. Let us put 



*=#!+</— i)f {h x t+h;p+ ,4H,r], (9i) 



the series stopping at the nth power of T and the coefficients 

 having the exact values given by equations (85), (86), and 

 (87). Then, substituting for </> in equation (51), 



T^t' + (/+ 1 - 2»«T) %- - (£ + 2im)f 



=2Jm(n + 0(/-l)fH„T". • (92) 



Now, equation (51) gives the value of </> caused by a 

 disturbing force represented by the two terms on the right- 

 hand side of the equation, and (92) is a similar equation, so 

 that we may regard the quantity on the right-hand side of 

 this equation as a disturbing force giving rise to fa just as 

 the disturbing force in (51) gives rise to <£. ]f the disturb- 

 ing force in (92) is small compared with the disturbing force 

 in (51), then it follows that fa will be small compared with 

 (f>, for physical considerations tell us that a small force must 

 produce a small effect. Since mT is very large compared 

 with unity, we need only consider the ratio of the disturbing- 

 force in (92) to the imaginary part of the disturbing force 

 in (51). This ratio will be least, of course, when the dis- 

 turbing force in (92) is least. If, by increasing n, we can 

 decrease the right-hand side of (92) we are sure to be 

 decreasing fa. Now the ratio of the disturbing force 

 affecting fa, when we carry the series to the nth. power of T, 

 to the disturbing force when we carry it to the (n — l)th 

 power, is 



n-j-f H n l n \-f 72th term of series 



n — r+ZH^T"- 1 n- I +/ (n-l)th term of series' 



n-\- f 



For all but small values of n the fraction rrr-/.is nearly 



n-l+f 



unity. This means that, as long as the terms in the series 

 for (jf> are decreasing, fa is becoming smaller as n increases, 

 and therefore the series approaches the value of cf>. This 



