60 REPORT— 1882. 



a second motion of tho same ellipsoid expressed by tlie equations 

 X = Ixq +.^l'yQ + ^l"za, ?/ = - mxQ + m'tj^ + - m"za, 



z—-nxQ +-n'yo + ?i'%, 

 a 



This law of reciprocity, applied to Jacobi's case, gives Dedekind's at once. 



The unknowns adopted by Dirichlet are not well adapted for consider- 

 ing the changes of shape and position of the fluid boundary ; what is 

 really wanted is the variation of the axes, their motion, and the motion of 

 the fluid relative to the ellipsoidal axes at any time. Riemann,^ taking 

 up the problem where left by Dirichlet and Dedekind, adopted as his 

 unknowns, a, the axes, their instantaneous rotations about themselves, 

 and the instantaneous rotations of a second set of axes, which give the 

 relative motion of the fluid, and which may be defined as follows. The 

 particles originally lying on the axes, at all future time lie on a set of 

 conjugate axes of the momentary ellipsoid — are, in fact, tho lines to which 

 the axes are deformed by a pure strain. If the momcntaiy ellipsoid be 

 changed by a pure strain to a sphere these become an orthogonal system, 

 and are the second system referred to. 



Having formed the difierential equations, and the integrals equivalent 

 to constant impulsive couple, the equation of energy, and the surface 

 integral of vortex strength, Riemann devotes his attention to considering 

 the general question of persistence of form, where therefore the axes 

 are constant, and for which his form of the equations is very suitable. 

 He proves that if the form is to be persistent the axis of rotation of the 

 fluid must lie in a principal plane of the ellipsoid, and must be fixed rela- 

 tively to it. Calling a h c the axes in descending order of magnitude we 

 may state his results as follows. For the more general case where tho 

 axis of rotation lies in a principal plane there are three sub-cases ; (a) 

 axis in plane of greatest and least axis, with a + c ^2b; (/3) axis also 



in same plane with « - c > 26 and c- < ^^, ~ f* ^ ; (y) axis in 



a^ — h^ 



plane of mean and greatest axis, with a — h > 2c and 



where B^ = fl + A^ A + M /'l + M. In this case there is neces- 



sary an external pressure in order to preserve the continuity of the fluid, 

 unless a, h, c are subject to another condition. In the more special case (^) 

 of motion about a principal axis, this axis must be either the least or the 

 mean. In fact calling c the axis of rotation and a the greatest axis, c 

 must lie between h + a and h — a' where a, a' depend on the solution of 

 a transcendental equation. If a decreases towards coincidence with I, 

 these limits for c become h and -303327 l, but if a = 6 exactly (Mac- 

 laurin's spheroid) c can have any value between o and a. Jacobi's and 

 Dedekind's motions belong to this case, which also serves to connect con- 

 tinuously cases (a) and (y), whilst case (/3) remains isolated. 



Riemann gives the following way of representing the foregoing per- 



1 'Ein Beitrag zn den Untersiichungen liber die Bewegiing eines fiussigen 

 gleichartigen Ellipsoides,' Ahh. lionifj. Ges. Wiss, Gott. Math. Class, ix. p. 3. 



