COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS, 399 
$7. Equal Roots of Equation (22) not Admissible.—At first sight it would appear 
that the conditions (33) and (40) of neutral stability would both be satisfied if 
equation (22) had equal roots: and real values can be found for P, T and w? which 
will satisfy this requirement. But if we merely substitute A, for A, (say) in the 
expressions (21) and (23) for # and y, we shall in effect be reducing the number of 
arbitrary constants involved in these expressions from eight to six, and it is not 
difficult to show that the boundary conditions can only be satisfied by making a and 
y vanish for all values of s: the shaft then remains straight, and, although this is a 
configuration of equilibrium, it is of no interest for present purposes. If, on the 
other hand, we write down the complete solution (with eight constants) under the 
condition of equal roots, we are in effect substituting A, for A, in expressions for « 
and y which differ from (21) and (23) in having sin A,s and cos A,s replaced by s cos 
A.s and s sin A,s respectively ; but when this latter modification is made in (21) and 
(23), and the criterion obtained by the same methods as before, it is found to be no 
longer satisfied by making A, equal to A,. 
"§8. Solution for ‘ Simply-supported’ Ends.—We proceed to interpret the solutions 
(33) and (40) in terms of the physical constants of our problem. Writing 
Al 
D) ’ 
# for we may throw equation (22) into the form 
HA—ApS—Bu?-C=6, . : : é : . (41) 
where A, B and C are the quantities defined in (1); and we may notice that two of 
the roots are necessarily real, since the constant term in this equation is negative. 
if we express the roots in the form 
My Mj=as db, \ : : ; ; : i (42) 
My My=e4d, | 
equation (41) must factorise as follows :— 
{u? —2ua +a*—b7} {u?—-2Que+ c?—d*} =O, 
and we have 
2(a+c)=A, 
a’ —b?+c*—d*+4ac= - 3B, 
ac? —d?) + e(a*—b) =0, G3) 
(a? —b*) (c?-d?) = —C. 
Again, the criterion (33) may be written as 
(H,°— 7) (My — my?) sin (Hy —- My) Sin (fy— My) | } , > 44) 
= (Hy? = 6) (Mo? — By) SID (MH, — My) Sin (My — 1), | 
and if we substitute from (42) this becomes 
l6abed sin (a+ b—c—d) sin (4a—b—c+d) 
= (a° + b*— 0? —d? + 2ab —2ed) (a? + b? — c? —d*— 2ab + 2cd) sin 2b sin 2d, 
= |8abed + (a*?— b*)* + (c?—d*)* — 2(a? + 6) (c? + d*)} sin 20 sin 2d, 
or 
8abed {cos 2b cos 2d—cos 2(a —c)} 
= {(@—b*)? + (c?—d*)?—2(a? + b*) (c? + d*)} sin 2bsin 2d... By eb) 
In this form, we can obviously deal with either real or imaginary values of b or d, 
observing that, if b= Bi, 
sin 2b _ sinh 28 
35 op? 
and 
cos 2b = cosh 28 
