Гидродинамика

SUMMARY 183 ma s s e s we r e b y equivalent solids The masses of the e qu i v a l e n t s oi d s are equal to those of the liquid. The centres of g r a v i t y о t V i e equivalent solids coincide with the centres of g r a v i t y of Ше l i q u i d masses. The moment of inertia of the equ i v a l en t s o l i d abou t any axis through its centre of gravity is shov^n t o b e than the moment of inertia about the same a x i s o f the c o r r e s p o n d i n g liquid mass (§ 9 and 10). When the b o d y h a s cavities in which for the liquid mas ­ s e s IS gfivena n i n i t i a l motion (i. e. are given initial circulations a r o u n d c l o s e d con t ou r s ) , then in replacing these masses by e q u i v a l e n t s o l i d s i t is necessary t o attach to the body a certain g y r o s c o p e , o f w h i c h t h e direction of the axis of rotation and t h e m o m e n t o f t h e initial momentum are completely determined b y t h e p r i n c i p i a l moment of momentum of t he liquid masses w h e n t h e b o d y - i s a t rest. The corresponding equations of mo­ t i o n a r e g i v e n b y formula (18) of § 10. Prof. Joukowsky points out t h a t in h i s me thod of treatment of the case of multiply- connec t ed c a v i t i e s there is no room for the doubt which was felt b y N e u m a n n in investigating the problem A t the e n d o f the chapter (§ 11) is given a brief outline of Neumann ' s m e t h o d of deducing the differential equations of mo t i on . The m e t h o d consists in applying Hamilton's Principle to t h e r i g id b o d y alone, which is considered as subjected to d i r e c t l y a p p l i e d forces and to liquid pressures. Prof. Joukowsky in h i s a n a l y s i s d e du c e s the differential equations of motion (§8, 9 a n d 10)b y f i r s t calculating the principal moments of mo ­ m e n t a of t h e b o d y ( § 9, formula 10) and of the liquid masses (§ 9 , f o rmu l a 1 2 ) , and then forming in the usual manner the d i f f e r ent i a l e q u a t i o n s of motion of the system with respect to a f i x e d point. Ch a p t e r I I d e a l s with the determination of the internal mo ­ t i o n of t h e l i q u i d and of t he ellipsoids of inertia of equivalent s o l i d s f o r d i f f e r e n t shapes of cavities. T l i e p o s s i b i l i t y of replacing a liquid mass by an equivalent solid was p r o v e d b y S t o l i i e s o n l y for the case when the velocities are low. Prof. Jou­ k o w s k y p r o v e s - t h e general case. - N e u m a n Hydi-odynamische Untersuchungeii, Leipzig, 1883, S. 236.