AND VIBRATION OF SHAFTS. 
295 
Case V. 
15. Shaft supported on three supports, and 4 feet apart respectively, a 
SUPPORT BEING AT EACH END. 
Thus— 
Fig. 8. 
Take the origin at A, and let dashed letters refer to the right of A, and undashed 
letters to the left. Then we have (§ 8, p. 287, equation 7) from C to A, 
y = A cosh mx + B sinh mx C cos mx + D sin mx, 
and from A to B, 
y' = A' cosh mx -f B' sinh mx + C' cos mx + B' sin mx. 
When X = 0 
y = 0, y' — 0, dyjdx — dy'/dx, d^yjdx^ = d'y'jdx^; 
when X — — Zj 
y = 0 , dhjjdx^ — 0 ; 
when X = 
y = 0 , dhj'ldx^ = 0 . 
Hence we get 
■A.+ C= 0.(1)^ 
A' H- C' = 0.(2), 
(B-B') + (D-D') = 0.(3), 
(A-A')-(C-C') = 0. (4), 
A cosh mil — B sinh ml^ + C cos ml-i — D sin ml-i = 0 . . . . (5), 
A cosh mil ~ ® mil ~ ^ cos mli d- D sin mli = 0 . . . . (6), 
A' cosh ml .2 + B' sinh ml^ + C' cos ml^ + D' sin ml^ = 0 . ( 7 ). 
A' cosh ml^ + B' sinh ml^ — C' cos ml .2 — D' sin ml^ = 0 . . . . (8). 
