430 Mr. R. V. Southwell on a Graphical Method 

 provided that 



20nH% ' ^ 



whence we obtain 



\= 9/5 X tt 4 = 175*36 approximately. 



To compare with this result, a solution was obtained 

 graphically. The length of the rod was divided into ten 

 parts, and after values of $8 and p had been calculated for 

 the points of sub-division, suitable lengths were chosen 



for bj, b 2 , b 3 , etc., and k l7 k 2 , k3, etc., as shown in 



the table below * : — 



Table I. 



x/l 



= 







o-i 



02 



0-3 



0-4 



05 



53 



= 



639 



682 



831 



1134 



1611 



1920 



P 



= 



•668 



•676 • 



•700 



•732 



•7595 



•770 



K(<x 1/p) 



cc 



1-498 



1-480 



1-429 



1-366 



1-317 



1-299' 



k (in inches) 



= 



5992 



5-920 



5-716 



5-464 



5-268 



5-196 



b (in inches) 



= 



3 195 



3-41 



4-155 



567 



8-055 



9-60 



A value for X was then guessed roughly, by estimating 

 mean values of ^ and p, and thence calculating the whirling- 

 speed from the ordinary formula. Substituting this value 

 for X in (18). it was found that the corresponding assump- 

 tion for the length OA in the diagrams would be about 

 5*71 inches, and this length was adopted at the first applica- 

 tion of the construction. Figs. 1 & 3 were then constructed 

 with initial slopes of 30°, as shown, for a first attempt, while 

 in the second attempt fig. 1 was started with zero slope, and 

 fig. 3 again with a slope of 30° : this second attempt gave 

 diagrams of which the general nature is indicated in the 

 left-hand bottom corner ot the Plate. The purpose of starting 

 fig. 1 with zero slope in the second attempt was to enable 

 the boundary conditions for M to be more easily satisfied by 

 synthesis, without altering the initial slope of the corre- 

 sponding Y-diagram : it is clear that the error will be pro- 

 portional to the initial slope, and hence, when constructing 



* The values here given for 33 and p are representative of a steel 

 shaft of diameter | in. at its terminal sections. For finding A, only the- 

 relative values of S3 and p at different sections are, of course, material. 



