778 Dr. J. R. Wilton on the Solution of 



Thus 



V = — 2irp(a 2 — b 2 — 2c 2 )^—co 2 (bc e~% cos rj + ab e~ 2 % cos 2rj 

 + ca e~^ cos 3?;) + kco 2 {(a + b)e~% cos rj + c e~ 2 ^ cos2tj}. 

 Hence 



^_^° = -2irp{d 2 -b*-2c i ) -co 2 {[bc-k{a + b)]e-^^ 

 + 2(ab-kc) e-W+^ + Zca e^^n)}, 

 and this must vanish at the singular points, where 



a-be- 2 V+^-2ce- 3 $ + ^=0. 



Consequently we have 



k=bc/(a + b), 4c 2 = a(a + Z>), 



(o 2 /27rp=(a + b)(a-2b)/3a 2 (7) 



Thus the curve 



x=(a+b) cost; 4- \a y(l-t-b/a) cos 2rj, 



y=(a — b) sin 77 — \a \/(l + b/a) sin 2rj, 



which is the three-cusped hypocycloid when b — 0, is a 

 possible form of rotating figure of equilibrium, provided that 

 o) is given by (7). But if b > the curve possesses loops, 

 and it is therefore not a proper solution, but must be regarded 

 as indicating that as the angular velocity diminishes the 

 fluid escapes at the cusps of the hypocycloid. 



More general cases of figures of equilibrium of this type 

 may be found without difficulty, but there is no great interest 

 in carrying on the investigation as all the figures so obtain- 

 able are, with the exception of the ellipse, unstable. 



Problems of Elastic Equilibrium, 



12. It is also possible to obtain solutions of certain problems 

 of elastic equilibrium, namely, the torsion problem*, the 

 flexure problem f, the problem of plane strain for a cylinder 

 bent by its own weight {, and the approximate theory of the 

 equilibrium' of a plane plate clamped or supported at the 

 edge§. But, except in those cases in which the solution is 

 well known, the analysis is tedious, and the results do not 

 appear to be of sufficient interest to repay the labour of 

 investigation. 



* Love, l Elasticity ' (second edition) p. 301, §§ 217-8. This is merely 

 the hydrodynamical problem of fluid in a rotating cylinder (with «= —1). 

 t Loc. cit. p. 317, § 229. J Loc. cit. p. 347, § 244. 



§ Loc. cit. p. 465, § 313. 



