FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
295 
quadratures are inappropriate for these integrations, because the function, say to 
be integrated increases so very rapidly. I therefore take an em^Dirical and integrable 
function, say v,,, which is such that ; the quadratures may then be 
applied to and the result applied as a correction to | v dip. In fact this 
correction is always very small, and we might well be satisfied to use | v dip, which is 
very easy to calculate. 
The empirical function v is given by 
V = u -^.2 e ^ “ 11 . 
Then when xjj = y, v = i when ip = y — 8, v = Uj^^; and 
i 
7 1 ^ 
V dip = 
log. 
^1 
qi - c ‘ “..j 
In all the cases I have to consider the exponential term is negligible, and the 
integral is . 
■'ll 
For the (juadratures we have 
^12 — ^^ 12 ) ^11 — ^ hi ) — ^ h 2 () 5 — '^12 
% 
V-nO 
, &C., 
and the equidistant values of the function, to be integrated (arranged backwards), are 
0, 0, 'Lo “'“n (~ M, Us — Uiol — ], &C. 
WJ WJ \yj 
The first two are zero, the next three or four are found to be sensible, and the rest 
are insensible ; hence the quadrature is very easy. 
The 33/ integrals are found thus :— 
13 / = 
iFi vo) «. + 5 cot" jS + G coU/3 + ... 
IP/ (2^0) 
The following table gives the 'll/, 33/ integrals • 
