36.2 Sir W. Thomson on the Motion of 



escape per second; and 424 kilogram-metres being required to 

 raise 1 kilog. of water 1° Cent., 424 x t kilogram- metres are re- 

 quired to raise 1 kilog. by t° ; it follows, therefore, that 424 X 

 t° x 1000 Q is the number of kilogram-metres necessary to raise 

 the water which escapes per 1" by t° ; 



.-. 424 x t°x 1000 Q = U 3 + U 4 , 

 or 



1 "424000Q' {qi0) 



in which expression Q, U 3 , and U 4 are given by equations (31) 

 (42), and (43). 



[To be continued.] 



XLVI. Hypokinetic Solutions and Observations. 

 By Sir William Thomson, F.R.S.* 



Part I. On the Motion of Free Solids through a Liquid. 



THIS paper commences with the following extract from the 

 author's private journal, of date January 6, 1858 : — 

 " Let f, ©, %, % flSL, $1 be rectangular components of an 

 impulsive force and an impulsive couple applied to a solid of in- 

 variable shape, with or without inertia of its own, in a perfect 

 liquid, and let u y v, w, ot, p, ar be the components of linear and 

 angular velocity generated. Then, if the vis vivaf (twice the 

 mechanical value) of the whole motion be, as it cannot but be, 

 given by the expression 



Q= [u, u]u 2 + [v, v]v 2 -f . • . + 2[#, u]vu + 2[w, u]wu + . . . 

 + 2[-cr, u\^u-\- . . ., 



where = [w, w], [v, v], &c. denote 21 constant coefficients de- 

 terminable by transcendental analysis from the form of the sur- 

 face of the solid, probably involving only elliptic transcendentals 

 when the surface is ellipsoidal : involving, of course, the moments 

 of inertia of the solid itself : we must have 



\u,u]u+ [y,u]v+ [w,u]w+ \yT } u]^T-j- [p,u]p + [cr,w]o-=^,&c, 

 [w,-57] u -f [v 3 'sr] v + [w,-5r] w -f [i3",'B r ] ot + [/0,-ar] p -f [cr^] cr = 31, &c. 



* Communicated by the Author. Parts I. and II. from the Proceedings 

 of the Royal Society of Edinburgh for 1870-71. Parts III. and IV. from 

 letters to Professor Tait of August 1871. Part V. appended to this com- 

 munication September 1871. 



t Henceforth T, instead of £Q, is used to denote the "mechanical 

 value," or, as it is now called, the " kinetic energy " of the motion. 



