A^^-D FIBRATIOX OF SHAFTS. 
287 
B = mass-moment of inertia about a diameter through its centre of gravity 
perpendicular to the axis of the shaft. 
HcncG 
Pt — L = £ 0 ^ (A — B) dyjd^c .(6). 
8. The solution to equation (3) is well known to bo 
y — K cosh mx -f- B sinh mx + C cos mx -f D sin mx . . . . (7). 
The quantities A, B, C, D are absolute constants between any two singular points, 
but have not necessarily the same values between every pair of singular points. 
If undashed symbols refer to those on the left, and dashed constants or symbols to 
those on the right of a singular point, then since the values oi y — dyjdx, are 
continuous, we have, at all singular points, whether points of supports or pulleys, 
y = y\ dyjd^ = dy'jdx ; 
whence 
(A — A') cosh mx + (B — B') sinh mx + (C — C') cos mx + (D — D') sin mx = 0 (8), 
(A — A') sinh mx + (B — B') cosh mx — (C — C') sin mx + (D — D') cos mx = 0 (9). 
But, at points of supports, y = 0, y' = 0 ; whence 
A cosh 7nx -j- B sinh 7nx + C cos mx -[- D sin 7nx =0 . . . (10), 
A' cosh 772X -f- B' sinh 77ix C' cos mx -f D' sin 77ix — 0 . . . (11 )• 
Also, since the bending moment is the same on both sides of a point of support, 
d^yjdx^ — dr'y'jdx^, whence 
(A — A') cosh mx -f- (B — B') sinh 7nx — (C — C') cos mx — (D — D') sin 7nx = 0 (12). 
At a singular point, consisting of a concentrated load, we have, from equations (2) 
and (5), 
(A, — A') sinh mx + (B — B') cosh mx -h (C — C') sin 7nx — (D -- D') cos 7nx 
W „ 
—-7-— w” (A cosh 7nx -I- B sinh mx + C cos mx 4- D sin 7nx] . . (1 3), 
and, from equations (2) and (6), 
(A — A') cosh mx + (B — B') sinh mx — (C — C') cos 7)ix — (D — D') sin mx 
= — —(A sinh 7nx + B cosh mx — C sin 7}ix + 1) cos mxj (14). 
