396 Sir William Thomson on the General Integration 



with this term or pair of terms. Hence we must have, for very 

 great values of i, 



K^l+l'H-l)', or K i= [7' + H- !)'>', ) 



k ■ ' V \ • ( 44 ) 



K f = — , or Kj= -jz, and so on. I 



P P l J 



Thus we see that if each of the roots p, p r , &c. is greater than 

 unity, the series (15) and (16) are necessarily convergent for all 

 values of /^ from pu = —1 to pu = 4-1, and they are divergent for 

 values of pu beyond these limits unless conditions proper to make 

 7=0, J' = 0, l n =0, F=0 arefulfilled. Butifone or more of the 

 roots p, p\ &c. is less than unity, and p the absolutely least of 

 them all, then unconditionally the series (15) and (16) are 

 necessarily convergent for all values of p> from — p to +p, 

 and they are divergent for all values of pu beyond these 

 limits unless a condition proper to make k = Q is fulfilled*. 

 When 7 = has imaginary roots, as a + /3\/ — -1, the absolute 

 magnitude of either of the pair is to be reckoned as (a 2 -f/3 2 )^ 

 and with this understanding the same statement as to conver- 

 gency and divergency holds as for real roots. But there is this 

 distinction in the circumstances of the loss of convergency in the 

 two cases, of transition through a real root and through the abso- 

 lute value of a pair of imaginary roots. In the latter case there 

 is no discontinuity when pu is continuously increased through the 

 critical value V (# 2 + /3 2 ) ; in the former, <£ and its differential 

 coefficients become infinite and imaginary, as pu is increased con- 

 tinuously up to and beyond any real root of 7 = 0. The inter- 

 pretation of the circumstances when imaginary roots of 7 = in- 

 fluence the solution is an exceedingly interesting subject, to which 

 I hope to return in a future communication. The remainder of 

 the present article must be confined to the case of y=0 having 

 two real roots, each less than unity. 



10. Let p be any real root of y = 0, and put p,=z + p. Then, 

 for infinitely small values of z } the differential equation (14) 

 becomes 



4( z f) +b * d i + ( c+d ^= e+ f*' ■ ■ < 45 ) 



* Mr.W. H. L. Russell, as I am informed by himself and Professor Cayley, 

 has given, in perfectly general terms, this criterion for the convergency of 

 the series in ascending powers of x for the integral of 



in a paper communicated to the Royal Society, of which certain extracts 

 have been published in the ' Proceedings ' for 1870, 1871, 1872. 



