OF ROTATIXG LIQUID CYLINDERS. 
99 
In figs. 5 and 6 the curves are those corresponding to ^ and 6^ = I. A glance 
at fig. 3 will show that there are difficulties in the way of drawing these latter curves 
with much accuracy. I have given in detail some of the calculations used in 
drawing the curve 6^ — 1, in order that the reader may judge for himself as to the 
closeness or otherwise of the approximations. The curve 0" = ^ is of course much 
easier. Before passing on to the calculations, two points ought to he noticed. 
(i.) It will be noticed that in the various d-series (the coefficients of powers of r in 
equation (128)) the terms last calculated are without exception of the same sign as 
those j^reviously calculated. There is therefore some justification for hoping that 
the remainders in these series will he of the same sign as these last terms. If this is 
so, the error introduced by the neglect of these remainders could, to an appreciable 
extent, he reduced by an adjustment in tlie value of 0. Thus we shall be attempting 
to calculate the curve for (say) d = 1, and shall obtain a curve which is much more 
like the curve for some smaller value of 6 (say 6 = '98) than it is like the curve 
6=1. Begarded as an attempt at tracing a surface of e<pnlibrium the error will be 
much smaller than if regarded as an attempt at tracing the particular curve 6=1, 
(ii.) It will be noticed that the sign of the leading terms in each of the series 
multiplying /•'b is positive. An examination of the method by which these 
leading terms are calculated will show that this is a general law ; all the leading 
terms after are of positive sign. Thus the error will be reduced by supposing the 
series (128) continued to higher powers of r by a suitably chosen series of terms. 
I have accordingly done this in the calculations, and the conjectural terms are, 
th roughout, enclosed in S(piare brackets."^ 
The Curve 0^ = (Fig. 5.) 
§ 30. In tracing this curve I started from (f, = n, and calculated a series of points 
on the curve for decreasing values of <^. The approximation at </> = tt was found to 
* The effect of these corrections must, of course, be small; but, at the same time, it seems as well to 
make use of any definite knowledge that we possess. 
