AND VIBRATION OF SHAFTS. 
333 
These experiments show that the formula used for calculating the resulting speed 
_viz., NiNg/.^/(Nj® + Ng^), where Nj, Ng are the separate speeds of whirl—holds, to 
a sufficient degree of approximation, whatever be the ratio of the spans or the position 
and size of the pulley. When one span is small compared to the other (Experiments 
119-125 and 98-104), the conditions approximate to those required in Case XIL, 
§35. In this case the error is sometimes positive and sometimes negative, and the 
percentage error, with one exception, is under 3. The average error is —'I. In 
Experiments 105-108 and 126-129, in which the ratio of the spans is one-fifth, the 
error is practically positive throughout. The mean error is -fi 3. The calculated 
results could be made to approximate more closely to the actual speeds by using the 
formula i^istead of N^Ng/v^ -fi Ng^), in which case the 
errors in Experiments 126, 127, 128, 129, 105, 106, 107, 108, would be — 2*0, —’5, 
— 3‘0, — 8'2, '4, O'O, 2‘8, O'O per cent, respectively, giving a mean error of — 1’3 per 
cent. But the speeds, as obtained by N^Ng/^/ -j- Ng®), are sufficiently near the 
actual speeds for practical purposes. 
When the spans are equal (Experiments 109-118)—which is the most important 
case—one span being loaded, the error is sometimes positive and sometimes negative, 
but in only two cases (Experiments 111, 117) does it slightly exceed three per. cent. 
The mean error is —•1 per cent. The experiments, therefore, amply verify the 
theory. 
45. The following are the mean results of experiments with the Pulleys I. and II. 
on the shaft at the same time. The spans have been each taken to be 16 inches. 
The first series include these experiments with Pulleys I. and II. on different spans, 
and the second with them on the same span—the positions in the two series being- 
similar. The notation used to determine the position of the pulleys is the follow¬ 
ing :—«!, ttg are the distances of Pulley I. and c^, of Pulley II. from the 
middle and outer bearings. The resulting calculated speed is taken to be 
NiN 2 Ng/Y/(N^®N 2 ^ fi- N 2 ®N 3 ^ + Ng^N^^), where N^, N 2 , Ng are the separate speeds of 
whirl for the shaft. Pulley I., and Pulley II. (see Case XVII., §§ 59-62 ; also §§ 30, 
34). The calculated speed for the two pulleys neglecting the shaft is given, having 
been calculated from the formula N^, = N 2 Ng/v/(N 2 ~ + Ng^). For the three causes 
together the resulting speed is NiN 4 ^/y/(Ni^ fi- N/), which is equivalent to 
N,N2Ng/v/(N,^N2^ + N2^XV + Ng^N,2). 
