I 
AND VIBRATION OF SHAFTS. 
343 
If, in addition, in the above equation we put (i) l.^ — ^ and (ii) oo^, it further reduces 
to the two equations already obtained in Case XII., § 35, p. 320, and Case X., § 26, 
p. 308. 
By making and each equal to infinity, the equation further reduces to 
«H^V(4 + Ci) + «[cp(4 + |ci) - 4- 4cj)} -4 = 0, 
which is the equation for a single span overhanging at one end and working in a 
shoulder at the other, the pulley being at the end of the overhanging portion. Thus 
Fig. 21. 
By putting, in this equation, ^ ^v’e obtain the equation already obtained in 
Case IX., §23, p. 305. 
If in equation [A] we put /g = oo , it immediately reduces to the equation already 
obtained for the case of a shaft of two spans, the shaft merely resting on the bearings 
at the ends, loaded with a pulley on one of the spans. (Case XIII., § 39, p. 325.) 
Again, if in equation [A] we make and each equal to infinity, it reduces to 
{(4 + 4) (4 + ^’i) + F 44} + « [ci^ {(4 + 
— F [(Zg + Zg) (Zg + 4(Ji) + ^ 
4) (4 + 3 ^i) + F 44} 
44}] (4 “h 4) ~ d. 
which is the equation for an overhanging shaft loaded at the end, and having two 
spans on one side. That is, for the case of 
