348 REPORTS ON THE STATE OF SCIENCE, ETC. 
The total magnitudes of the component couples which act at any seetion are, 
dy 
therefore, " 
aa E zy aa 
=e I be 
ds ds* ds re ds* fe 
These component couples represent actions produced by the stresses across the 
section P upon that part of the shaft which lies between O and P. In addition, 
the stresses will produce component forces, in the direction of the axes Oz, Oy, 
Oz, which we may represent by X, Y and Z. Now the component accelerations, 
due to rotation, of an element of the shaft situated at P are 
teh an, and wy and O, 
q g 
in the directions Ow, Oy, Oz respectively; hence, the equations of equilibrium 
for the element are 
Ay 
= += Ow =O}, 2 : : 4 . (4) 
io iz oty’ = Opto 1) be: SHEER ites Tawtieny 
2 = OR bY ~~ -+ eR TGS ty 
o (ae — wie) oS Yk ge 30... 
o (vd 4 wits) +X - 20" =O. S RINNE eee eas 
and & x xe + 34 So. - > )- eee 
From (4) and (5) it is evident that X and Y are small quantities of the first 
order. Hence, neglecting small quantities of the second order in (9), we may write 
aT’_ 9, 
ds 
whence 
T’ = const, = T (by end conditions). F 3 ; . (10) 
We have similarly, from (6), 
% = const. = —P (by end conditions) ; : ; , - Gay 
and on substituting from (10) and (11) in (7) and (8), differentiating, and 
aY 
substituting for Ss and - ae from (4) and (5), we obtain, finally, as the equations of 
neutral stability, 
a (de wily, ,W . ad*y 
Soff ld Meester LN hilo EF peasy | tec jg : 4 4 12 
dst\. ds = g ne ds’ oe 
and 
ad* (dy a, W d*z i 
SY | eset) (eee WS a Pase= 0: A . f F 13 
= ds bs = q ab g* Cs 
These two equations replace the six equations (4)-(9). 
§5. Special Case of Zero Torque.—When T is zero, a solution of (13) may be 
obtained by assuming that «=O identically. Equation (12) then reduces to 
d'y d?y - Ww" 
pas! eg Pact ei iO Alas < . 3 eG 
ds* x ds* g y Soe) 
and a solution is obtainable in the form 
EI 
y =K, sin as + K, cos as + K, sinh s+ K, cosh Bs, . . . (45) 
