OF ROTATING LIQUID CYIANDERS. 
69 
Since we have sii|)posed 
equation (I), the equation 
equation (2) to represent the complete solution of 
^ = F (£) 
( 8 ) 
must necessarily represent the complete solution of equation (6), if the meaning 
of F remains unaltered. Let us suppose that 1l:e values of the multiple-valued 
function F (^) (or, what is the same thing, F [x + ^y)) are exhibited on the appro¬ 
priate PtiEMANN’s surface. The locus of points at which this function is equal to 
X — iy (also a definite and unique function of position upon this Riemaiqn’s surface) 
will be the complete system of points satisfying equation (6), and will therefore 
include the curve S. I’his curve will not, howeAmr, be draAvn upon a plane, but 
upon a Rieaiann’s surface. 
Now so long as the curve S does not possess a cusp or branch point we have at 
eA'ery point of S, 
<'!• If) 
(I (.1 — i//) 
'U 
d (.>■ •+ iji) 
and hence it fidlows that no branch point of the PiIEMANn’s surface can lie on the 
curve S. Let the curAm S be a single closed cui'Am surrounding the origin, and it 
folloAA^s that Ave can ahvays so arrange the Piemann’s surface that no branch line 
shall intersect the curAm vS. In this case it is possible for a .solution of the form of 
equation (6) to satisfy at eA’ery point of the curve S, the curve uoaa'- l)eing regarded 
as a closed curve lying on one sheet of a Riemann’s surface. By simply Interchanging 
the axes of y and — y Ave can imagine the fVmction F (t;) represented on the same 
Riemann’s surface, and Ave see tliat at eAmry point on the old curve S Ave shall haAm 
f = F(t7) = 4>{y) + xlj{r]). 
We have therefore proAmd that if a jiarticular solution of the foim of equation (3) 
can be found to represent the curAm S, then solution (7) aauII rejiresent the same 
curve. We haAm also seen that if a family of curves is described by continuous 
deformation starting from some cui'Am Sq, the general solution can 1)6 arriAmd at by 
continuous variation starting from the solution for the curve 50 long as no 
member of the family ymssesses a cusp or branch point. For our present purpose 
it is sufficient to haA’e proAmd that if AA-e haAm, at every point of the surface S, 
^ z= (P (y) xjj {f . (9), 
then Ave haAm also, at every jioint, 
y = <p{^) ii) 
Let us now introduce a new function y defined by 
X = 0 + (pi f) cU 
^ (f) + 4> iv) — I rp {y) dy 
j -0 R 
?y 
AAdiere C is a constant. 
We haAm by differentiation, 
^ (0 + ^ • • ( 12 ), dxfy = (p {y) + xp {v) — ^ . 
and therefore at the surface S, by equations (9) and (10), 9 x/ 9 ^ = ^x/^y = 0. 
( 10 ). 
(II). 
(13). 
