EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION, 939 



Hence, when the real part of /3 is positive and j ^8 1 is large, approximately 



D,„=4/3'^J(J2 + J'>^) + 8^5J'(J2 + J'2)-2/3y{6m2J2 + (4m2 + y)J'2}, . . • (4) 



= 8 J27r-3/?s 



'sin a + 1 { Icos 3a + (1«.2 + ^) cos a } + -1 { - (1. + Am^ + ^) sin 3a + r sin a | 



,(5) 



where the value of r need not be written down. 



The two highest terms in (4) being equivalent to 



4/35(/3.T + 2J')(J2 + J'2), (6) 



we may infer that the high roots of D,„ are approximately the same as those of 

 ^J + 2J' (all real), and of P + J'^ (all complex). 

 For m = 0, D^ reduces rigorously to 



2/33(./3J + 2J'){2/32(J2 + j'2)_^j'2}^ . . . . . . (7) 



but for any other integral value of m, D,„ is irreducible. 



Asymptotic expressions for the zeroes of D,„ are easily found from (5). 



Real roots. 



We see that (5) changes sign between Nvri^, if N is sufficiently large. 



For the root between those limits we have the approximate equations 



sill a = 0, a=N7r + e, 



SO that 



Complex roots. 

 For these 



a sin a + — COS 3a = 0, 

 4 



or, if a = ^ + i>], where ^ and »] are large, real and positive, 



4i(| + iv) + (cos 2| - i sin 2^)e2') = 0, 

 leading to 



621 sin 2^ -4^ = 0. I 

 Thus cos 2^ is small and positive, and 



then 



621 = 4^. 



Hence 



/3_8^ = a = N7r + ^ + i--Llog4N7r, (10) 



a sufficient approximation for the applications given below. 

 More approximately, 



Number of zeroes of D,,, ivithin given limits. 



By an application of the Theory of Functions of a Complex Variable, I have found 



