428 M. G. It. Dahlander on the Form assumed by a Fluid 



for the innermost bounding surface is x 2 -\-y 9 + z <i =r i , and 

 xdx -f ydy -f- zdz = 0. Hence we find that 



w*-M + M'=-N + N' . . . (3) 

 is a conditional equation, and equation (2) will therefore be 



(_ N + N'-^7 + \*pf v ^ f+ ^d* + y dy + 2 d:)=Q. (4) 



The surfaces de niveau are therefore all spherical. If their 

 radius be denoted by r 1 ) we get 



xdx + ydy-\-zdz=r J d? J ; (5) 



and if the pressure be denoted by p, we get 



^=(-N+N'-|v/+|^)^- • (6) 

 By integrating this we find 



J - (-N + NO j - 1 -*w*- \ v/$ + c 



But ^ = r when p = 0, whence 



J.(_N + NO £=3- 1 V/V 2 -'- 2 )- 1 Tf^(p-l)- (7) 



Equation (7) gives the pressure which takes place in dif- 

 ferent parts of the fluid. If r 1 be taken as the radius R of the 

 outer sphere, then p must be =0. Thus we get the conditional 

 equation, 



(-n+no £!=£ U|«^it_^ + 1 v^(i-e)=o. (8) 



Equations (3) and (8) determine the relation which must exist 

 between the quantities therein occurring,in order that equilibrium 



may be possible under the given conditions. If |r be denoted by 



k, then equation (8) becomes 



^_N_fN'_3 v/ ^ (1 + A) + 4 v/ ^ =a > {9) 



Hence we find that if, under the given conditions as to the 

 dimensions of the spheroid and the density of the fluid, equili- 

 brium is possible in one case, there are an infinite number of 

 other cases in which equilibrium can also take place, provided r 

 and R may vary but Xr^is supposed to be constant. 



Equation (9) is of the second degree with regard to k ; but 



