OF ROTATING LIQUID CYLINDERS. 'J7 
inquire, are the values of r and 6 over which this approximation may be regarded 
as good ? 
The coefficient of each power of r is an infinite series of powers of 6, of which 
all terms up to have been calculated. A glance at these series will show that the 
approximation is tolerably good so long as 6~ <C 1, but begins to break down as soon 
as 6 exceeds this unit value. 
Supposing that we have assigned to d some definite value less than unity, the 
value of tfi (r, 0) will be given by an infinite series of powers of r, of which only 
the first seven are known. For small values of r these first few terms will give a 
sufficiently good approximation, for larger values the approximation will he bad. 
e 
and for still greater values the series will become divergent, so that the first few 
terms give no approximation at all. It will be seen from inspection of equation (128) 
that the approximation will be toleralfiy good so long as < \j6^. 
The conditions under which the calculated terms will give a good approximation 
may accordingly be supposed to be that 6^ < and < i/^b In fig. 3 is 
represented the plane of r, 6 . The part of this plane over which the approximation 
is good is that hounded hy the four curves 
d = 1, 6* = — 1, r9 = I, r9 = — 1. 
This is the portion which is shaded in the figure. 
VOL. CO.— 
A. 
o 
