954 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



and' 



|8(z-/) 



fz-2 P}£Z±J „ ^2 S /,'• a»-l 





^Zl^P/^J i 



Hence 





^ 



(3) 



u., 



.<mjd(3 \ 13 



I ?^P(2) + 5|. ^P(z)+ .... to the term in a""' or a" \ 



TT _ ^(^-Zl) 



+2 



1 fi^^^^)-*-;8^aT>^+----+^aiP^(^^M 



e a 



.d'DJdl3 

 ' U 







(4) 



cZD,„./cZ/3 I ' ' 13" jz, dz" ' ' p 



Consider separately the four series in the four lines on the right of (4). 

 In the_^7"5^ line, the sum of the series can be found in finite terms, for it is 



Residue at (3 of ^ \ ^V{z) + -^^- ^P(z) + .... to the term in a'-^ or a" [ . . (5) 



The function of which the residue is taken here is an odd uniform function of ^ 

 which vanishes at infinity in such a way that the sum of all its residues vanishes. But 

 the residues at ±:/3 are equal, so that 



2[ ] in (5) = - — residue at of the function of /S in [ ] 



- J-coefficient of Ji in ^'^ | ^^P(z) + 

 2 B T>^\ B ^^ 



P 



2«3 ^ 



{z)+ . . 



to the term in a"~^ or a" >. 



(6) 



u. 



The ascending power expansion of ^r has the form 



^ = u!:^'^-* + uir^'iS-^ + ur^s"" + u^'^^ + 



(7) 



where U^ *^ and U^ ^^ do not occur if m>l. 



m m 



(Note that the forms of U^\ U^\ etc., are diflferent in the two cases p>p', f><p', 

 Art. 16.) 



Hence in (4) the 



Jig 



first line of I2 = - aU™P(z) - a^Ul"' -2P(z) - .... to the term in ft"-i or a". . . (8) 



In next article this will be considered more fully in connection with the terms 

 found in Art. 38. 



In the second line of (4), the general term is proportional to the displacement 



Up = JJ,„e a (9) 



belonging to one of the transitory free modes. 



