OF EOTATING LIQUID CYLINDEES. 
83 
The value of oy is given by 
I — ca^l'Iirp = ijn — l) 
It therefore appears that the angular velocity decreases as we recede from the point 
of bifurcation. 
Throughout the series the l^ounding curve is periodic in 6 with a period 27r/r/, and 
it is easily seen that at the remote end of the series is a curve consisting of n equal 
and symmetrically arranged arms, these arms extending in the limit to infinity, and 
being of zero breadth at all points except in the immediate neighbourhood of the 
origin. 
In fig. I I have drawn the curve for the case of n = 3, error in 
the value of Y, (ecpiation (38)) caused I)V the neglect of will l3e seen to be of the 
.,.G 
order of '000377^ — cos 4d. 
Since the maximum value of r*^'/cd on tins curve is 8o®, it appears to be legitimate 
to neolect this error. 
O 
Points of Bifurcation on Elliptic Series. 
§ 19. Let us now return to the series of elliptic cylinders, and search for points of 
bifurcation on this series. We have found in § 13 that when cq is finite, but when 
the sc[uares and products of the remaining n’s may be neglected, we have (equation 
(44)) for values of n different from 2, 
f„ (« 1 , an, 03 ,...) = ——r—- — -{- terms linear in o„+ 4 ,, &c. . . (61). 
L ^ccCt ;^ 7Z 
Let us now examine the form assumed by equation (54) at points upon this series. 
We have, in the first place, dfjda:, = 0. 
M 2 
