292 
MR. S. DUNKERLBY ON THE WHIRLING 
The s^eneral solution to equation [A] is best obtained by assuming c = al, where a 
is less than unity, and expanding each term in ascending powers of ml. In this 
manner we get, to a sufficient degree of approximation, the equation 
From this equation the following results—giving the values of ml for different 
values of a —have been obtained. 
Ratio a. 
Value of ml. 
Unity. 
1-506 
Three-quarters . . 
1-902 
One-half. 
2-507 
One-third. 
2-905 
One-quarter .... 
.3-009 
One-fifth. 
3-044 
One-sixth. 
3-060 
One-seventh .... 
3-069 
One-eighth .... 
3-071 
One-ninth .... 
3-073 
One-tenth .... 
3-078 
Very small .... 
3-080 
If we assume ml — A, then the number of revolutions will be a maximum for a 
given length [I + c) of shafting, when A(1 + a) is a maximum. From the above 
results the speed will be a maximum when the ratio (a) is one-third. 
Hence, for a shaft of given length running on two bearings, one being placed at 
the end, the best position for the other bearing is such that it divides the length of 
the shafting in the proportion of 1 : 3. 
In all cases that occur in practice the overhanging portion is small compared to the 
span. Hence, we may say that if a shaft, span I, overhang a distance less tlian 
one-fifth the span, then ml = 3’07 8. 
13. Experimental Results. 
The following are the mean results, the calculated speeds being obtained according 
to the formulm in the preceding article (p. 292), when the particular value of c/l is 
taken. 
