940 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



the result that when the positive integer N is large, the number of zeroes of D,„ whose 

 real part lies between 



±{(N4> + 'f-^} 

 is GN + 3m+6. 



The result is probably true down to N=l. and even, except when 7U = 0, down 

 to N = 0. 



To find the number of zeroes with positive real part less than a given value, we 

 have to take account of the number of zeroes at 0, and as to this the cases of m = and 

 m= 1 have to be considered specially. For to>1, m= 1, w = the number of zeroes 

 at is 3m +2, 9, 8 respectively. (Arts. 11, 12, 13.) 



Hence, for m > 1, the number of zeroes with positive real part less than 



TYlTT TT 



(^4) 



, TT + - - _ is 3N + 2 

 2/2 4 



1 TT 



for m= 1, the number with positive real part less than (N+— )7r + -is3N; 



and for m = 0, the number with positive real part less than N7r + - is 3N — 1. 



No purely imaginary zeroes. 



That D,„ has no purely imaginary zeroes follows easily from physical considerations. 

 For if any such existed, there would be a real solution periodic in z with no surface 

 traction or body force. The potential energy between any two z-planes at an interval 

 of a period would vanish, since the work done by the end-tractions on the end-displace- 

 ments would consist of two parts equal except for sign. 



Except /3 = 0, the roots appear to be all simple. For the large roots this is easily 

 seen from the approximate value of dDJd^ at a zero of D^. If D^ had a multiple 

 root, the forms of the residues at that root would of course be altered, but the method 

 of solution would still hold good, and the transitory character of the resulting functions 

 still be maintained. 



31. Approximate formula for the high transitory mode due to a unit force at 

 the surface or in the body. 

 Real zero ofJ)^. z'^z'. 

 By 30 (2), (8), for a real zero of D,„, 



/3-8„ = a = N7r, (1) 



ttV N\a/ (a a ) 



(2) 



4(v+2)m2 ^ 2(v + 2)2m2 ^ 



^1=--^^. ^i=- \J , Ci = 4m(v + 2)N, 



A2 = - 8mN, Bg = - 4?n(v + 2)N, C^ = SWir^ 



A3 = 8m2N, B3 = 4m2(„ + 2)N, C3=-8mN37r2, 



^ 



(3) 



