FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
293 
the values v/hich would be found froiu “ Harnioiiics ” without this correction, and if 
(f).-,, (jf)/ are the required values, it appears that 
/O /O ? 
where a, e, a', e are the coefficients specified in ^ 12 of “ Harmonics.” 
The approximate values found in this way for all the (f> integrals are very near to 
the more correct values, and might have been adopted throughout without material 
error. But as there was not much certainty that the apjoroximation was a good one 
—and indeed for Sq was probably bad—I also found all these integrals, excepting 
<pQ^, by the method now to Ije descriijed. 
From (6) and (8) it appears that 
TT Sin- 7 
(10 rl(f). 
If, therefore, we vuite 
= u^AL ■ 
— 2a6xV.2„ ffi (2ac‘ -|- h') A.],, — ['2(td + '2bc) A®,; + . . . 
+ 2«7>'fl|, + (2a'c' + h'-) ni„ + (2« d' + 21/0') ffi . . . 
for = — 1,0. 
we have 
6 
TT Sin- 7 
The following table gives the results for all the o)/ p/ <i/ integrals :— 
Table of Logarithms of (jy, co, p Integrals. 
?. 
b*. 
log </>A 
Aj)pi’oximate log 
from “ Harmonics.” 
log < 0 ;'+ 10. 
log pi^+ 10. 
0 
0 
— 
— 
— 
7-63099 
2 
0 
9-0051G-10 
9-00515 - 10 
7-G7.371 
7-02716 
2 
2 
7-0.3970 - 10 
7-0397.3 - 10 
(-) 5-G819.3 
(-) 5-05256 
4 
0 
9-09080- 10 
9-G932.3 - 10 
8-03.358 
7-35625 
4 
2 
1-72G64 
1-72729 
(-) 8-33.367 
(-) 7-59132 
4 
4 
.3-81.541 
.3-81G12 
8-29058 
7-37446 
G 
0 
9-71219-10 
9-G9.305- 10 
7-97301 
7-32602 
G 
2 
— 
2 -205G2 
(-) 8-72778 
(-) 7-91094 
G 
4 
— 
5-29999 
9-10094 
8-13161 
