352 REPORTS ON THE STATE OF SCIENCE, ETC. 
shown that the condition (32) may be written in the form 
(AP — Ax?) (Ag? -AY*) Cos (Ay + AQ -Ay— ie 
— AL —Ag?) (AP AY) COS (Ay + Ag Ay ry 
<i 5 A l ie 
— (A> = Ag*) (AF—A4*) cos (A, Ta = O, . . (83) 
which may therefore be taken as the criterion of neutral stability for ‘simply 
supported ’ ends. 
The Criterion for ‘Clamped Ends.’—On the assumption that the bearings prevent 
both displacement and change of direction, the terminal conditions take the form 
o_O; 
=0,| 
de -when s= + Be : : : - (34) 
ae) , 
ds 
Substituting for # and y in these equations from (21) and (23), we obtain the 
relations 
, 1 : : 
A, sina, 5 +A, sin et A, sin A, : + A,sinaA, =0, : . (85) 
1 Y Z , U ne 
A, cos A, 5+ A, cosa, 5 + A, cosa, 5+ A,cosa, 5 =O, ? . (86) 
, l ; l : L i U ‘ 
AjA, Sim Ay | + ALA, Sin A, > + AgAy SIN A, |G +A,A,SIOA,S=O0, . (37) 
1 
A,A, cos Ay) + AgAy cos A, : 9 t Ass 008 Ay UF AA, CO9A, | =O, - (38) 
and a similar set of four relations, which may be obtained by writing B,, B., . 
for Aj, A,, ... inthe above. Hither set of four relations yields the same equation 
in 7, if we eliminate the coefficients A,, A,,...or B, B.,... ete. and the 
condition for neutral stability may therefore be written in the form 
l Z 1 
sin A, 2, sin A, 5, sin A, ~ 9? sin A, mm 
1 l 1 Z 
cos A, 2 Cos A, ,, cos Ay, COS Ay 5, 
e 3 7 =O... - (39) 
: Z Z Z , 1 
A, sin A, 9" A, sin A, 3° A, Sin A, - 53 A, sin A,.,, 
A, COSA : A, A, : A A, : A A : 
1 1 9? cos 29° 3 COS Ay 5, , COS A, 9? 
Expanding the determinant, we obtain, as the criterion in this instance, the 
equation 
(A, —AyQ) (Ag—Ayq) COS (Ay +AL— Ag —A,) 
—(Az—Ag) (Ay—Aq) COS (Ag+ Ag—Ay—A) 5 
eae 
— (A, —Ag) (A, —Ay) Cos (Ay +A, —A,—A,) 5 =O, . : - (40) 
in which A,, A,, Ay, A, are, as before, the roots of equation (22). 
