AND VIBRATION OF SHAFTS. 
359 
where </>o, ^3 are constants depending on the position and size of the pulleys ; and 
(assuming all the pulleys to be so near together that each affects the others), the 
formula 
__ 
+ Ng^Nd + NdNd) 
(A) 
will only be correct provided 
Even when these relations do not hold, it is shown in §§ 29, 30, 33, 34, 44 and 45, 
that the formula (A) is, with certain modifications and restrictions to suit particular 
cases (§§ 33, 34, 45), sufficiently accurate for practical purposes. 
The formula (A) may be extended to any number of disturbing elements. If, for 
example, there were four, and the speeds corresponding to them be Nj, Ng, N 3 , 
then the resulting whirling speed is 
__ 
v/(Nd~NdNd + NdNg^Nd + + N/N^Nd) ’ 
and so on. 
Considering the case of two disturbing elements, if their speeds of whirling, taken 
separately, be each equal to N, the resulting whirling speed due to two causes com¬ 
bined is N/y/ 2 . 
If there were three disturbing elements, and if their speeds of whirling were all 
equal to N, the resulting whirling speed would be N/y^3. 
Of two disturbing elements, if the whirling speed for one of them be four times 
that of the other, the resulting whirling speed is not more than three per cent, less 
than the smaller whirling speed. 
Concluding Remaids. 
63. In conclusion, it should be noticed that in finding the speed at which a 
continuous shaft of given diameter, supported on bearings placed at equal distances 
apart, and loaded with pulleys on any or all of the spans, will whirl, the first step is to 
find the span which will have the biggest whirl (that is to say, the span which 
carries the heaviest and most advantageously situated pulleys as regards whirling), 
and to consider this span and the spans immediately adjacent to it on either side. 
The span in question can, in general, be determined on, at a glance, from the 
consideration of the weights, sizes, and positions of the pulleys which each span 
carries. Having fixed upon the three spans, the next step is to find (by the formula 
for each case) the whirling speed for the shaft and each of the pulleys on the three 
spans in question, on the assumption that the effect of every cause, except the one 
