148 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



is a member of the confocal family, say that for which f= - 

 Comparing with Art. 100 (2) we see that if a, c be the polar and 

 equatorial radii, and e the eccentricity of the meridian section we 

 must have 



The surface condition is given by Art. 96 (1), viz. we must 

 have 



^ = -iu& 2 (l-^ 2 )(f 2 -l) + const ............. (1), 



for = f - Hence putting n= 1 in Art. 101 (14), and introducing 

 an arbitrary multiplier A, we have 



- ...... (2), 



with the condition 



bo ~ 



1 - e 2 2e 3 1 

 The corresponding formula for the velocity-potential is 



(4). 



The kinetic energy, and thence the inertia-coefficient due to 

 the fluid, may be readily calculated, if required, by the formula (5) 

 of Art. 93. 



103. Leaving the case of symmetry, the solutions of V 2 $ = 

 when $ is a tesseral or sectorial harmonic in /j, and w are found by 

 a similar method to be of the types 



where, as in Art. 87, 



T n s (M) = (1 ffi s /; (3), 



x/x * / rt i S \ f* 



