346 REPORTS ON THE STATE OF SCIENCE, ETC. 
at the ends. The co-ordinates actually employe’ are non-dimensional quantities, 
A, B and C, related to the dimensions and material of the shaft, and to P, w and T, 
thus :— 
eee 
“oer | 
ugh? 
~ ZBI [aed . : p : 5 “eel 
and C= Wat}, | 
16gE1 
where 
1 denotes the length of the shaft between bearings, 
EI denotes the flexural rigidity of the shaft, 
and 5 denotes the mass of the shaft per unit length. 
Any self-consistent system of units may be employed. 
The diagrams have been extended into the region in which B is negative, for the 
purpose of illustrating the effect of end tension in maintaining stability. Negative 
values of C are of course inadmissible, since they correspond to imaginary values of 
P| | | | AAANATE TT the numbers attached 
- ed eds] ok NN fo esac aay Valucs46-0 
7-0 
NBS A20 WS | 
Tog CAN “aia soul 
MEEREEVEANN (SSN CeaeCaIE 
AEA 
1-0 | f 
FECEREPRE RNASE 
“50 -40 -30 -20 -l0 ° "0 20 30 40 50 
Values of 46 : 
Fig. 18. 
w, and the sign of A (i.e., of the torque) is immaterial. In fig. 17 dotted lines have 
been drawn to represent, for values of A corresponding to the calculated contours, 
the contours which would be given by the formula 
(=y+ 4B 4 16C_y P ; .Q 
ae are 
This formula can be shown to be exact when either A or C is zero, and the close 
agreement between the two sets of contours shows it to be a very close approximation 
in all cases where B is positive. A similar formula, with the terms on the left hand 
side altered by the addition of coefficients 0°49, 0:25 and 0°195 respectively, represents, 
with rather less close approximation, the results of fig. 18. 
§3. Perhaps the most striking feature of the problem is the relative complexity of 
the analysis which leads to such simple results. Although shown by actual com- 
parison to be closely equivalent to a formula of the type (2), the exact relation 
obtaining in either case between A, B and C is of a highly transcendental nature, 
and its interpretation entails much laborious calculation. For practical purposes, 
the diagrams will be quite sufficiently accurate, and more convenient than 
3 The authors’ thanks are due to Miss S. W. Skan for her valuable assistance in 
this work. 
