Free Solids through a Liquid. 363 



If now a continuous force X, Y, Z, and a continuous couple 

 L, M, N, referred to axes fixed in the body, is applied, and if 

 f, . . . , &c. denote the impulsive force and couple capable of gene- 

 rating from rest the motion u, v, w, -cr, p, cr, which exists in 

 reality at any time t ; or, merely mathematically, if f &c. denote 

 for brevity the preceding linear functions of the components of 

 motion, the equations of motion are as follow : — 



f-f. + X P =X, §= & c, 



^ .... (1) 



at 

 ^-fv + 9u-%p + &* = -$. .. 



Three first integrals, when 



X=0, Y = 0, Z = 0, L = 0, M = 0, N=0; . (2) 

 must of course be, and obviously are, 



f 2 -fg 2 + Z 2 = const (3) 



resultant momentum constant ; 



£? + #&» + #%= const (4) 



resultant of moment of momentum constant ; and 



u? + i& + wZ + iA + pm + <r& = Q," . . (5) 



These equations were communicated in a letter to Professor 

 Stokes, of date (probably January) 1858, and they were referred 

 to by Professor Rankine, in his first paper on Stream-lines, 

 communicated to the Royal Society of London*, July 1863. 



They are now communicated to the Royal Society of Edin- 

 burgh, and the following proof is added : — 



Let P be any point fixed relatively to the body; and at time t, 

 let its coordinates relatively to axes OX, OY, OZ, fixed in space, 

 be a, y, z. Let PA, PB, PC be three rectangular axes fixed re- 



* These equations will be very conveniently called the Eulerian equa- 

 tions of the motion. They correspond precisely to Euler's equations for 

 the rotation of a rigid body, and include them as a particular case. As 

 Euler seems to have been the first to give equations of motion in terms of 

 coordinate components of velocity and force referred to lines fixed relatively 

 to the moving body, it will be not only convenient, but just, to designate 

 as " Eulerian equations " any equations of motion in which the lines of 

 reference, whether for position, or velocity, or moment of momentum, or 

 force, or couple, move with the body, or the bodies, whose motion is the 

 subject. 



