EECENT PEOflPvESS IN HYDRODYNAMICS. 59 



in general tlie same. The coefficients of the original co-ordinates will be 

 nine in number, and functions of the time alone. Substituting the 

 velocities and the attractive forces in the equations of motion, it is found 

 that the initial co-ordinates enter linearly, and hence, in order to have a 

 free surface, the pressure must be of the form 



P + ,, A _ ^ _ ^' _ V>^ 



where the coefficient of a equated to zero gives the initial surface, and o- 

 is a function of the time alone. Equating to zero the coefficients of 

 ^Oi 2/0. ^0 there result, with the equation of continuity, ten equations to 

 determine ff and the nine coefficients. This is the fundamental idea • 

 for the development I must refer the reader to the paper itself, contenting 

 myself here with giving some of the chief physical results of Dirichlet's 

 investigation. This was confined, so far as he worked it out in detail, to 

 surfaces of revolution. When there is no rotation, and the original form 

 is an oblate or prolate spheroid at rest, the form vibrates through the 

 sphere to a prolate or oblate spheroid respectively, and he finds equations 

 to determine the limits and the time of vibration. If in any position the 

 velocity of change of an axis surpasses a certain limit the form does not 

 vibrate, but the spheroid either lengthens infinitely or flattens infinitely, 

 but the presence of the slightest amount of rotation prevents the former 

 ultimate state, a result easily foreseen from the constancy of ano-ular 

 momentum. When rotation occurs three cases present themselves, dis- 

 tinguished by the relation of the angular velocity to the momentary form. 

 The first gives no change of form, and leads to Maclaurin's spheroid ; in 

 the second the spheroid vibrates as well as rotates ; and in the third it 

 rotates and either flattens itself without limit, or, in the reverse direction 

 tends to an ultimate form not of infinite length. In the second case the 

 motion is only possible without a uniform external pressure over the 

 surface, provided the angular velocity at the moment of greatest 

 lengthening is less than a certain limit. The last section of this paper is 

 nearly all due to Dedekind. In the foregoing the same particles always 

 form the principal axes of the ellipsoid. Dedekind states here that there 

 are only two other cases in which this is the case, one in which an ellip- 

 soid vibrates without rotation, for which the co-ordinates are propor- 

 tionate to their initial values, and the other is Jacobi's ellipsoid. He 

 further states another possible motion where an ellipsoid satisfying 

 Jacobi's conditions retains its form stationary in space, with an internal 

 motion of the particles given by 



X — Xq cos ht + J jIq sin Id, y = —xq- sin Id + t/q cos M, z = Zq. 



U (J, 



This may be referred to as Dedekind's ellipsoid. 



The proof of these theorems Dedekind gave in an appendix ' to 

 Du-ichlet's paper republished in Borchardt's Journal, and in addition a 

 remarkable reciprocal law between two correlated motions with the same 

 boundary surface. It is thus stated by him. To every motion of a fluid 

 ellipsoid expressed by the equations x = Itiq + myo + nzo, y = I'xq +m'yo 

 + n';s„, z = l"x(, + m"yo + n"zQ whose original surface has the equation 



'-Y +-J^ + 2 ~^ corresponds, by changing the initial state of motion, 

 ' ' Zusatz zu der yorstehenden Abhandlmig,' Borch. Iviii, p. 217. 



