1882.] 



Liquid Ellipsoid about an Axis. 



211 



leading (by the preceding paper) to the equations 



[x 2 - (2a 2 - b 2 - c 2 ) fi + (c 2 + a 2 ) (a 2 + b 2 ) - 4« 4 = 0, 

 y? - (2b 2 - c 2 -a 2 )/ji + (a 2 + h ! ) (6 s + c 2 ) - W = 0, 

 p* _ (2 C 2 - a 2 - b 2 ) fi + (b 2 + c 2 ) (c 2 + a 2 ) - 4c 4 = 0. 



These three equations cannot co-exist, and therefore we must put 

 one of the three H t , n. 2 , fl 3 ; and therefore also one of the three 

 f , T), £ equal to zero. 



Suppose 12, = 0, and therefore £ = 0; then 



fi 2 - (2a 2 - b 2 - c 2 ) fi + (c 2 + a 2 ) (a 2 + b 2 ) - 4a 4 = 0. 



b 2 c 2 



Puttinsf —y = x, -5=v; then the roots of this quadratic in a 

 6 a" o 1 * 



are real if (x — yf — 8 (x + y) + 16 is positive ; and therefore if the 

 point (x, y) lie on the shaded part of the diagram, AL being a 

 parabola, focus S, vertex A, and OL = 4 ; supposing also b 2 > c 2 , or 

 x> y, which can be done without loss of generality. 



Fig. 1. 



The ellipsoid is now a surface of equal pressure if 

 a* A' = b 2 B' = c 2 C ; 



a . 4c'V(c 2 -a 2 ) „ 

 a 2 A + -V- V 



c — a 



- 1 aV 



_ 4a'6'(a'-&' ) _ / a 2 -6" + ^^ 

 /a" V M / 



= 6'£ + 4a ' 6 '<"'- 6 V -f^^'-l)V 



^ 2 \ fJb • / 



., . , 4cV (c a - a 2 ) , fc 2 -a 2 V" ,, , 



