498 Lateral Vibration of Bars supported at Two Points. 
For given ratios c/l we can obtain values o£ ml from the 
above equation. This has been done incidentally by Professor 
Dunkerley in connexion with his work on the whirling of 
shafts *, but the calculations were not sufficiently extended to 
give very accurate results. By comparison with the table 
given below, it will be seen that the approximate formula 
used by Dunkerley cannot be relied on to give more than 
the first two significant figures. 
Having recently required more accurate solutions, I have 
found it necessary to make a more elaborate calculation, and 
as the results have been obtained to six figures it appears 
desirable to place them on record. 
If we write a for c/l and 6 for ml, and expand each term 
of the equation in ascending powers of 6 and aO, we get, as 
far as the twenty-first powers, 
46)2 _ (. 4 + .4 a s + .3 ^5,6 + (-000035273369 + '0021164 a 3 
+ -0037 a 4 + '0021164 a 7 + '00079365 a 8 )6> 10 - ('58730 + 
106-8890 a 3 + 293'945 a A + 1007-811 a 1 + 881*834 a 8 + 
106-889 a 11 + 26'722 a 12 )10- 8 6> 14 + ('033 + 13'2 a 3 + 49 a 4 
+ 508-8 a 7 + 700 a 8 + 509 a 11 + 297 a 12 + 13'2 a 15 + 2*4 a 16 )lO- n 18 
= 0. 
Assuming values of a' and calculating 6 we obtain Jthe 
numbers tabulated below: — 
Ratio c/l. 
Value of 9. 
Unity 
1-50592 
1-90170 
2-51895 
2-94042 
3-05881 
3-(.9975 
3-11752 
3-12647 
3-13148 
3-13449 
3-13641 
3-14159 
Three-quarters 
One-half 
One- third 
One-quarter .... 
One-fifth 
One-sixth 
One-seventh .... 
One-eighth 
One-ninth 
One-tenth 
Zero 
Phil. Trans. A, 1894, p. 279. 
