58 EEPORT— 1881. 



and shows bow they are simplified when the body has three planes of 

 symmetry. For this case he finds the generalised equations of motion in 

 terms of the pressures exerted on the surface of the solid, and considers 

 separately pure translation and pure rotation. Amongst results, be 

 notices that the fluid moves in ' stream lines ' (in Faden), that the effect 

 of the fluid is to cause an apparent increase of mass, different in different 

 directions, a diminution of gravity ; and states the following laws : — ■ 

 * The form of the curves on which the particles of the fluid appear to 

 move relatively to the centre of gravity of the body depends only on the 

 form of the body and the curve on Avhich its centre of gravity moves.' 

 These results had already, several years before, been given by Stokes.' 



In the next year appeared another paper, by the same aufhor. On a 

 general transformation of the liydrodynaviical equations."^ This is ex- 

 ceedingly general, and starts with the consideration of equations analogous 

 to the hydrodynamical equations in n variables, viz. : — 



dY c?M, dui dui dui 



dxi dt dxi dxi " dx„ 



&c., together with 



dui _^ du^ ^ + ^ = (1) 



fZis, dx2 dx„ 



He takes n arbitrary functions Uq, a^, . , . . a„_i, of the variables, 



forms their Jacobian, and takes the minors of —-5, -— , Calling 



ax I dx2 

 these Ai, Ao, . . . , they satisfy the equation 



dAi d^2 _ .. 



a.'i'i dx^ 



The transformation now consists in making «,...(/„_] the dependent 

 variables, being connected with the old ones by the equations Mj = Aj, &c., 

 which is possible, since the equation (1) is satisfied. Introducing the 

 quantity T, defined by the equation 2T = A^j + . . + A^^, he proves 

 that the above equations are equivalent to others of the form 



/ (V-T)=A^^' +K,^' + . . . +^^' (2) 



ax I ax I dxi dt 



where A, . . . are functions of «[ . . . a„_ j. If the motion is steady, 

 these produce 



V-T = r(a, . . . a„_0, A,„=|l' 



da,„ 



The investigation is necessarily complicated, and the reader is referred, 

 for the full working out, to the paper itself. He shows, amongst other 

 things, that when the motion is steady, the equations are the conditions 



that ... (T — ¥)dxi . . . dx,^is a maximum or minimum. Intro- 

 ducing the condition u^ = -—1, &c., he shows that for steady motion, 



the integrals are a^ ^ const : &c., and that tliese give the lines of motion. 

 After reducing the foregoing results to the case of three variables, he shows 

 how to transform the independent variables to others. Lastly, he applies 



' ' On some cases of fluid motion,' Camb. Phil. Trans., viii. ' Crelle, liv. 



