RECENT PKOGtRESS IN HYDRODYNAMICS. V3 



alone, Lagrange's equations are applicable directly to tlio expression for 

 the energy. Now in this, the coefficients are, in general, functions of the 

 angular co-ordinates of the body referred to fixed axes. If the energy 

 be expressed in terms of the velocities referred to axes fixed in the body, 

 the expression has constant coefficients. It is therefore advantageous 

 to determine the form of the equations of motion when the energy is so 

 expressed. This had been effected by him in 1858,' but was not pub- 

 lished until 1871.^ At the same time also, the form of the equation for 

 the motion of a single solid, with any number of apertures in it and 

 cyclic motion through them, a case in which the conditions of direct 

 applicability of Lagi-ange's equations are not satisfied, were deduced. 

 When there are several solids, of which some at least have apertures, the 

 equations are naturally more complicated. The equations of motion for 

 this case were published^ by the same author in March, 1872 ; they have 

 also been proved in a different manner in the new edition of Thomson 

 and Tait's ' Natural Philosophy,' and have been already referred to. 



I adduce here some of the chief results of the above analysis as 

 developed by Sir W. Thomson. In the ' Natural Philosophy ' was con- 

 sidered the case of the motion of a solid of revolution in an infinite fluid 

 so as always to keep its axis in one plane. A certain fixed point in the 

 axis (the centre of reaction) determinate when the form and distribution 

 of mass in the body is known, moves in a sinuous line cutting the line of 

 resultant impulse at equal intervals, and the body swings about the 

 centre of reaction according to the law of the quadrantal pendulum. If 

 A, B are the impulses necessary to produce unit velocity along and per- 

 pendicular to the axis, and jx the impulsive couple to produce unit rotation, 

 then the length of the simple pendulum keeping time with the swinging 

 of the body is gnAB/E,^(A — B), ^ being the resultant impulsive force of 

 the motion. This is on the supposition that there is no perforation with 

 cyclic motion through it. An example of this latter was solved in the 

 same manner for a circular tore with no rotation round its axis, in the 

 paper ' On the motion of free solids through a liquid.' Here, when the 

 ring is projected with a rotation round a diameter, its axis oscillates 

 rotationally according to the law of a horizontal magnetic needle carry- 

 ing a bar of soft iron rigidly attached to it parallel to the magnetic axis. 

 In the same paper Thomson has also treated the question of the general 

 motion of a solid of ' complete isotropy with helicoidal quality.'* The 

 point P about which the body is isotropic moves uniformly in a circle or 

 spiral so as to keep at a constant distance from the axis of the impulse, 

 and the components of angular velocity round any three rectangular axes 

 are constant. 



In the ' vortex motion ' he has shown, from general reasoning, that if 

 a solid moves from a great distance past a fixed obstacle to a great 



' Referred to by Eankine (1863) ' On plane water lines in two dimensions,' Trans. 

 Itoy_. Sue. (1864). Reprint of Rankine's Papers, p. 495. 



= ' On the motion of free solids through a liquid,' Proc. Jt.S. Edinhurgh, vii. 

 p. 384. See also p. 60. 



' ' On the motion of rigid solids in a liquid circulating irrotationally through per- 

 forations in them or in any fixed solid,' Ihid. p. 668. 



_ * Tlie following is Thomson's example of such an isotropic helicoid : • Take a 

 uniform sphere and place on it projecting vanes in the proper positions— f.^., cutting 

 at 45° each at the middles of the twelve quadrants of any three mutually perpen- 

 dicular great circles.' Of course the vanes in practice must not have sharp angles, 

 else discontinuous motions will mask the effects. 



