1886.] I Equilibrium for a Rotating Mass of Fluid. 333 



then by trial and error to find two values of a. one degree apart, one 

 of which made the left-hand side of (28) positive, and the other 

 negative. 



The smallest value of 7 is 54° 21' 27", but after that value integral 

 degrees for 7 were always chosen. 



The solutions for 7 = 55° and 57° could not be found very exactly 

 from the elliptic integrals with logarithms of only seven figures, but 

 the solutions were confirmed by the approximate formulae (21) and 

 (22). The solution for 7=80° was confirmed by the approximate 

 formulas (23) and (24), and that for 7 =85° was only computed 

 therefrom, since when 7=80° the approximate formula gave nearly 

 identical results with the exact one. 



The solution obtained is embodied in the table opposite. The first 

 three columns give the auxiliary angles 7, «*, from which the remain- 

 ing results are computed. 



As a graphical result is much more intelligible than a numerical 

 one, I have given two figures, showing the three principal sections in 

 two cases, namely, where 7=60°, and 7= 75°. For these figures a is. 

 taken as 2 cm., so that the volume of fluid is |tt x 2 3 cubic cm. 



