Ml;. .1. II. JEANS UN THE VIEEATIONS AND 
170 
lo obtain approximate iiimierical solutions, my jMan has been to draw graphs of 
the functions and in tliis way obtain a graphical solution of the equations for 
dilierent values ol p. Tliere is no difficulty in drawing graplis of the functions ; 
tliese are trigonometrical functions, and we have 
— ~ — cot .r, 
wliile the successive ilh are connected hv the relation 
■hi + 1 
1 
To save space 1 have suppressed all details of this somewhat tedious part of the 
work, file results lor u = 1, 2, 3 are given in tlie following table :— 
Lowest Values of x. 
P = o. 
P = U. 
/7. = A. 
71= 1 
4-49 
4-2 
4-0 
n = 2 
5-76 
5-6 
5-4 
ti='S 
6-98 
6-8 
6-7 
lor large values of n it will he found that equation (54) reduces approximatelv to 
aq = 0, and hence that for any value of p tlie lowest value of x is slightly less than 
the corresponding value in the case in which p = 0. 
The First Point of Bifurcation. 
^20. It therefore appears that the first point of bifurcation may he safely assumed 
to he of order n - 1. The value of a; for which it occurs y\i\\ liave some value 
hffiween 4*0 and 4-49, according to the value of p/X. Now =: kg, and the value of 
K IS Airp /(X “1 Hence the first point of liifurcation is ajiproxnnately given by 
■iirp-a- _ = IG‘00, Avheii p = X, 
X + i> |4-493 = 20-lG, Avhen p r= 0. 
In equation (3) we sup|)osed this point of bifurcation to he given by 
ypycT/X^ = <f)- 
In oni present analysis Ave have already taken y = 1 ; if AA'e take (X -f 2p) to he 
Identical Avith our former X^, AVe see that the actual Audues of (X ai’e rom''hlA" 
ij) = DGU, Avhen p = 0, if = 1-27, Avhen p = X.=^^ 
* It Avill be fuuud that the first iioiiit of Mfurcatioii is given, Avitli great aeeuraev, by the single 
equation + = 1-6 for all value.? of /x between 0 and A. This is of interest,^ls shoAving the 
relative importance of A and /. in maintaining stability. As might be foreseen, the importance of /r 
relatnxdy to A increases as n increases, and for n = co, the factor A + -'/x must be replaced by A + 2/-. 
