OF EOTATING LIQUID CYLINDERS. 
75 
least equal to 1 /r. This function is real at every point of space, and is a solution of 
Laplace’s equation. Hence the function is finite at every point of space. 
It is therefore clear that if, in equations (16) and (17), we replace ^{yj) by 
(f) [t], c), xjj (jj, c), and make the corresponding changes in </> (^), ifj (^), we shall have a 
solution which satisfies all the conditions of the problem, subject to the single 
condition that c is less than Cj. At infinity xf/ {rj, c) will be capable of expansion in 
the form of equation (5), 
xjj {rj, c) = b^/r) + 62 / 7^2 + + ... , 
and we now see that equations ( 21 ) and ( 22 ) can be obtained in the same manner as 
before. 
It is therefore clear that the values obtained for V/, A, and a will he the true 
values, even if they have been obtained by the use of divergent series, provided only 
that the series for remains convergent up to the boundary. 
In the case in which cq < c.^, a similar proposition is true for values of c such that 
G C.i. 
§ 10 . Y\ e now consider the case in which c is greater than either or Co. Let tlie 
values of V^- and Vq which have Ijeen found for values of c less than either or c.^ he 
extended, by a process of “ continuation,” to points outside their circles of convergence, 
and let the values so obtained define the functions V,- and Vq. These functions will 
have certain infinities, the position of these infinities depending upon the value of c. 
When c = 0 , all the infinities of V; lie outside S ; all the infinities of V,, lie inside S. 
Let us suppose that up to some value of c, say c = cq, no infinity crosses S, l)ut that 
(if possilde) at c = cq one of these infinities is found on the boundary. The values of 
V^- and Vq are functions of c, and V; — Vq satisfies the requisite algebraical ecpiations 
from c = 0 until c is equal to the smaller of the values c = or c.,. Hence it must 
continue to satisfy for all values of c, until the condition found in § 3 is violated, i.e., 
until the value of c is such that the curve possesses a cusp or branch point. Also V, 
and Vq satisfy the requisite conditions of finiteness, uniqueness, and continuity until 
c reaches the value C 3 . Now as c approximates to Cg from the direction in which 
c < cq, the value of V at same point of the boundary (viz., the point at which the 
infinity occurs when c = Cg) will increase indefinitely, becoming ultimately infinite 
when c = cq. This value of V will, however, give the true solution for all values of c 
less than Cg, and there will be a superior finite limit to the value of V. It therefore 
follows that there can be no value cq at which an infinity crosses the boundary, and 
the values of V; and Vq found by ‘‘ continuation” of the power series will give the 
true values of Vj and Vq until the whole solution is invalidated by tlie occurrence of 
a cusp or branch point. 
Summing up, it appears that we may neglect the question of convergency of series 
altogether ; so long as the values obtained for either V^ or Vq are possible values, 
they must he true values. But care must be taken not to pass through values of the 
parameter such that S possesses a cusp or branch point for these values. 
L 2 
