mi. J. H. JEANS ON THE EQUIEHiPHlEM 
7 G 
§ 11. Illustrations of these remarks are aftbrded by the examples of §§ G and 7. In 
equations (23), (24), and (25) let ns regard c as a variable parameter, so that, as c 
varies, tlie equations represent the dilferent members of a family of circular cylinders. 
We have seen that the series obtained for ^ only remains convergent so long as c <|-a. 
Equations (26) and (27) represent the values of Yj and Vg expanded in powers of c. 
Now the value of V,- is convergent, no matter how great c may be, hence we know 
that this represents the true value of Vi for all values of c. The value of Vg has as 
its circle of convergence the circle r = c, and this intersects the boundary as soon as 
c attains the value c = ^a. Hence the value obtained for Vg will fail to give a true 
solution as soon as c exceeds the value c = la, although it will always give the value 
of Vg at points outside the circle r = c. Again, in § 7, let us regard y/ 5 as a variable 
parameter. The value found for V, is convergent for all values of ^76, and therefore 
represents tlie true solution for all values of provided that we start from a true 
sohitlon, and do not pass through a value of at which a cusp occurs. Under this 
same condition the series for Vg will give the true value of Vg at all points at which 
it is convergent, and the expression given in equation (32) will give the true value at 
all points, this being the expression for Vg which would be found by “ continuation” 
of the series, or (what is the same thing) by the methods of § 9. 
The ellipse does not possess a cusp except in the critical cases of = 0, ^/h = oo , 
In the former the ellipse reduces to a pair of parallel lines, and the points at infinity 
rank as cusps. In the latter the ellipse reduces to a doubled straight line joining the 
])oints X — ^ a~^, and, again, these points rank as cusps. Hence a solution will 
remain the true solution so long as ^^b does not pass througli either of the values 
.^b = 0 or CO, i.c., so long as y^b does not change sign. This is the meaning of the 
condition found in § 7, that ^^/a and ^Jb must be taken with the same sign. 
We can see this from another point of view, as follows. The solution (29) can be 
exhibited on a Riemaxx’s surface of two sheets, the branch points being given by 
abif =zb — a. 
When a =• b (?'.c., when the ellipse reduces to a circle) these points coincide in the 
oi'igln, and destroy one another. As b increases, the two points move along the axis 
of ,r, and ultimately meet the curve when b = co at the points a’ = i i.e., meet 
tlie cuiwe at its cusps as soon as cusps occur. Similarly, as b decreases from the value 
b — a, tlie branch points move along the axis of y, and meet the curve when 5 = 0 
at the points y = co , 
Deformed Circular Cylinder. 
§ 12. As an example of exjiansion in a series of powers of a parameter, let us 
consider the cylinder of wliicli the equation is 
m — cr -f- 2cr’' cos n9 
or, in y coordinates, 
= «' + C (^""^ -{- y’‘) 
(35), 
(3C). 
