350 REPORTS ON THE STATE OF SCIENCE, ETC. 
§6. General Case.—Reverting to equations (12) and (13), and eliminating a or Y; 
we obtain the equation 
» a as za aN \4 
[2 5+ (EI pe EV 2 yO. See ae 
ds§ dsi* ds* y°) 4 oe 
The solution may be written in the form 
(@, y)=2[A, sin A,s+B, cos A,s], 
where A,,A,,.  *. ete., are the roots of the equation 
(BIA — Pat— War) —TA®= 0. 
g 
Writing this equation in the form 
Blat—Pa?— Va? = a Tas, 
g 
we see that the roots obtained by taking the negative sign on the right-hand side 
are equal and opposite in sign to those which are obtained by taking the positive 
sign. The most general expression for w is therefore given by 
w=A, sin A\s+A, sin A,s+ A, sin As + A, sin A,s 
+ P, cos A\s+ B, cos A,s +B, cos Ays + By cos A4s, ; : . (21) 
where A,, A,, A,, A, are roots of the equation 
; sh hee 
EIA‘— Ta?— Pa?— — a? =0; 1 SAR 
g 
and the corresponding expression for y can be obtained from (13) in the form 
y =B, sin A\s + B, sin A,s + B, sin A,s + B, sin Ays 
—A, cos A\s—A, cos A,s— A, cos Ays — A, Cos Ays. 4 : . (23) 
These expressions are legitimate whether A,, A,, A, A, be real, imaginary, or 
complex. 
The Criterion for ‘ Simply Supported’ Ends.—On the assumptions (a) that the 
journals prevent movement of the ends of the shaft, but without constraining the 
direction-of the bent centre-line, and (b) that the twisting couple applied at the 
ends of the shaft has its axis parallel to Oz, the terminal conditions may be expressed 
as follows :— 
Za), 
a O, 
dy das eae 
LS +E. =O, \ when s= +5 (24) 
da d*y } 
TT —EIl_” =0, 
ds ds? 
The last two equations represent the condition that at the ends the resultant couple 
on the shaft has Oz for its axis. 
If we substitute for 2 and y from (21) and (23), these conditions yield the 
relations 
A, sin rs +A,sina,! +A, sin ee +Aysind=0,. «ea 
1 ma Aes Taher eel aa 
A, COSA, | + A, Cos A, 5 +A,COSA,> + 4008 Ay = Ho : > ae 
(BIA,—Ta,) A, sin a, 6 + (EIA? TA,) Agsin A, : 
+ (Ela, —'A,) A, sin Ab + (BIAS = TAA, sina, =0,.. 4 se 
Jie 
