1892.] Maclaurins Liquid Spheroid, 33 



11. We shall now apply (15) to investigate the stability of 

 a Maclaurin's spheroid wliich is composed of viscous liquid. 



Let the original surface be confocal with the spheroid 

 and let the system of confocal hyperboloids be 



and let p = y be the free surface of the original figure. 



Let ds^, ds^ be line elements measured along the normals of 

 the spheroid and hyperboloid, and p the central perpendicular on 

 to the tangent planes to \he former, then 



ds^ = h'^dv, ds^ = h~^dfi, 

 also 



whence 



/,^ = ^, h,= P^-^, (16), 



and 



dS = c'y (1 + rf)p-'diidy\r, dn = ds^ = c^yp-'dv (17). 



The values of U^ , U^ may be written 



U, = tA-p,r (7) q:' {v)P:' (/^) cos m^, 

 U, = lA- q- {ry)p- (v) P- iix) cos m^. 

 Accordingly from (14) and (17), 



a = £-^^ tA:^ {p'-q- -p:4T) P: {h^ cos m^. 



From the formulae given by Mr Bryan*, 



n — n 

 n-{- m\ 



^1 -^ '77) T 



«'« —Pn ' "'" V ) « j_ ^iy» 



^ J- m\ (—V"' 

 whence K"'?;" -i^.^g?' =^^^^-^ ^-^, , 



and therefore 



S (_)- J.;» 'UL^: p;» (^) cos «^./r..(18). 



47rcV (1 + 7') " ' ' "" -^^-w! 



* Pi-oc. Roy. Soc, vol. xlvii. p. 368. 

 VOL. VIII. PT. I. 



