COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 351 
(BIA2—TA,) A, cos A, 4 + (WIA, TA,) A, cos Ay 4 
+ (BIA,?—TA,) Ay cos Ay 4 + (EIAZ—TA,)A, cos a, 4 = 0, 
and a similar set of four relations, which may be obtained by writing B,, B,,.. . 
for A,, A,,.... in the above. Hither set of four relations obviously yields the 
same equation in J, if we eliminate the coefficients A,, A,,... or B, B,... . 
Subtracting (25), multiplied by P, from (27), and remembering that, by (22), 
MTR a, OT oy kd Go Ag ener he aon 
g ™ 
when A has any of the values A,, A,, Aj, Ay, We see that for (27) we may sutstitute 
the relation 
AG es G WAG tA, tA, 1 
be: : “4 « = ; = on "=O 2 
a? sin A, 2 + A? sin A, 2 ae sin A, 5 2 sin A, 5 OP . (80) 
Similarly, for (28) we may substitute the relation 
A, (Plas (jes-ood ss Te l 
A? cos A, a*h2 cos A, ata? COS A, ata? cos A, Sao c . (381) 
Then from (25), (26), (30) and (31), eliminating A,, A,, A,, A,, we have, as the 
required criterion for a condition of neutral stability in the free-ended shaft, 
sin A, “ sin A, sin A, a sin A, 4 
cos A, . cos Aas) COS Ay , cos A ie 
=O 32 
sin A, Ss iti — sin A, : "sin A, : = 
2 2 A? "2 A; Br mong 2’ 
ut Cus A, an 4 cos A, oh D cos A, by by cos A, hy 
A? D aaeea 2 ay? 2 a? 2 
The exparsion of the determinant on the left-hand side of (32) is 
l 
coseca, -. fe Gh ppt Bec of (ieee (fae 1 
2 | (A?—A,*) (Ay?—A,) sim A, 5 sin (A, — As) 7oo (A,-Ay) 5 
Aj? Ay? Ay Ay? — A 2 
— (A\2—A,”) (A.2—A,?) sin Ag 5 sin (A, —A,) i sin (A, - A,) i 
a - 
L 
+ (A,?—A4?) (A,?—A,”) sin A, — sin (A,—A,) 5 sin (A, - A,) 2) 
wun 
and since 
(A\2—A42) (Ay? — Ag?) = (AY? = Ag?) (An? Ag?) — (APY?) (AY? A?), 
and the quantity 
1 
A? ies Ag? Ay? 
gl 
—which, by (22). is equal to Wa 
») —cannot, by hypothesis, be zero, it is easily 
