AND VIBRATION OP SHAFTS. 
291 
When X — —1, 
1/ = 0, d^yjdx^ — 0, 
and when a? = 0, 
?/ = 0, ]f — 0, dyjdx = dy/dx, d^yjdx^ = d'^y'/dx^. 
Also, when x = c, 
d~y'Idx' = 0, d^y'/dx^ = 0 (shearing force zero). 
Hence, we get 
A cosh ml — B sinh ml -f C cos ml — T) sin ml = 0 , . . . (1), 
A cosh ml — B sinh ml — C cos ml + D sin ml = 0 . . . . (2), 
A + C = 0.(3), 
A'+C' = 0.(4), 
(B-B') + (D-D') = 0.. . . (5), 
(A-A')-(C - C') = 0.. . .- (6), 
A' cosh me + B' sinh me — C' cos me — D' sin me = 0 . . . . (7), 
A' sinh me + B' cosh me -j- C' sin me — D' cos me = 0 . . . . (8). 
The elimination of A : B : C : D : A' : B': C' : D' from these equations leads to the 
result 
(cosh ml sin ml — sinh ml cos ml) X (cosh me sin me — sinh me cos me) 
— 2 sinh ml sin ml {\ + cosh me cos me) = 0.[A]. 
If Z = 0, by dividing throughout by sinh ml sin ml, the equation reduces to 
1 + cosh me cos me = 0 
the equation already obtained for an overhanging shaft fixed in direction at one end 
(Case 1, § 9, p. 289). 
If c = 0, the equation [A] reduces to sinh ml sin ml = 0, i.e., sin ml = 0, the 
equation already obtained for a shaft resting freely on a bearing at each end (Case II, 
§ 10, p. 290). 
2 P 2 
