AND VIBRATION OF SHAFTS. 
335 
The formula by which the resulting speed is calculated, viz. :— 
N,N,N3/v/(N/N/ 4- + N32N,2), 
gives, of course, the same calculated speed whether the pulleys be on different spans, 
or similarly placed on the same span. The experiments show that, with the pulleys 
on different spans, the observed speed is higher (with one exception) than when pulleys 
are similarly placed on the same span. In Experiments 138 and 148 the observed 
speed is the same in each case. Moreover, with the pulleys on different spans, the 
observed speed is, with one or two excej)tions, in excess of the calculated speed ; 
whilst, when on the same span it is, without exception, less than the calculated speed. 
In the former case, the average error is about + 3 per cent., and in the latter, about 
— 5 per cent., giving a mean of — 1 per cent. Either one or other of the separate 
errors (Experiments 130-140 or 141-149) could be reduced by the introduction of a 
Constantin the denominator of the expression NjN 3 N 3 /.y/(N^^N 2 ^ + N/N 3 ^ + Ng^N^^), 
as in §§ 33, 34, but whilst reducing one it would also increase the other. 
Considering, hoivever, the complexity of the problem the preceding residts justify^ 
to a remarkable degree, the assumptions that have had to be made in the course of the 
investigation. 
The experiments made with the pulleys on different spans are very instructive as 
showing how one pulley affects the other in regard to whirling. For example, 
Experiments 130, 131, 135, and 134 show that, when the two pulleys are both taken 
into account, the calculated speed is much too low. Hence we may Infer that if 
Pulley I (which is the lighter of the two) be placed on the far side of the centre of 
its span from the middle bearing, its effect on the whirling speed is very small. The 
whirling speed may, in fact, be taken as that resulting from the combined effects 
of the heavier pulley and the shaft. On this assumption, the calculated whirling 
speeds in the above four experiments would be (see § 44, Experiments 110, 109) 3056, 
3056, 2600 and 2600 respectively, and the percentage errors would be -f- 5, + 6’2, 
+ 7, and + 3’4, instead of + 4‘4, -|- 8'7, + 4'7, and + 6’4. 
46. The discrepancies between the observed and calculated results are accounted 
for by the fact that the empirical formula—viz., N^N 2 /.y/(N^^ + N^^)—upon which the 
resulting calculated speeds are based, is not strictly accurate. In the case, however, 
of two or more equal spans with pulleys on each span, that formula gives calculated 
results less than the observed results and, therefore, erring on the right side. This 
is apparent from Experiment 130-140, but it might also be proved by considering the 
case of two equal spans with a pulley placed in the centre of each span. If the two 
pulleys be of the same size and weight they will have, separately, the same whirling 
speed. Let that whirling speed be N^. Then, from § 41 or 42, we have 
cc 2‘900 v/(^EI/WZ®), about, 
where I = length of a single span. Using the ordinaiy formula, the resulting 
