730 
PROFESSOR W. M. HICKS OK THE THEORY" OF YORTEX RINGS. 
We shall assume—an assumption to be justified by the result—that 
(A +B„T n )/(A„_ l Tt„_ 1 -f B„_ 1 T„_ 1 ) 
is of order k, or of the first order of small quantities by which the approximation 
proceeds. In this case A„/A*_ 1 is of order k z , and B^/B^ of zero order, in other 
words A„/A 0 of order k~ >l , and B„ of the same order as B 0 , but there is no means at 
present of determining the order of A 0 , or A' 0 with reference to B 0 . 
2. The most general case considered is where the core is hollow, and has therefore 
an inner free surface, and an outer surface common with the fluid moving irrotation- 
ally. The cross sections of these surfaces will approximate to circles. Take the 
critical circle (radius a) of our curvilinear coordinates to be that belonging to the 
mean circle of the inner cross section, and for this mean circle let k be k v Its 
actual form can then be represented by the equation 
k=k 1 (l+a, cos 2-y+ a 3 cos 3i>+ . . . ).(3) 
where a n is of order k n . 
The outer surface can be represented by the form 
k=k 2 (l-]-/3 1 cos v-\-/3 2 cos 2v-\-/3 s cos 3v-f . . . ).(4) 
where, as in the former case, /3„ is of order k n . 
It might be thought that it would be possible to represent the outer surface by an 
equation of the first form, and the inner surface by one of the second form. But we 
have no right to do this, for the inner surface might not contain the critical circle, and 
it would then be impossible to represent it by the second form. To assume that form 
for it would, therefore, be equivalent to assuming that the inner surface contains the 
critical circle of the outer. Now, the outer must evidently contain that of the inner, 
hence the equations above given can actually represent them. 
The mean circle approximating most closely to the outer surface will not be that 
represented by k 2 , but one which does not belong to the system k at all. It will be 
necessary to know the distance of its centre from that of the inner mean circle. 
Now [T. F., Eq. 6] if R, r, denote the radius of the axial circle, and of the cross- 
section respectively of a tore (u), 
ItyV=C a/r— S 
Hence for the inner surface 
radius of cross-section=a/S =2ak l 
radius of ring =mO/S = a(l + 2£ 1 3 ) 
}to second order. 
