76 EEPORT— 1881. 



■where A is a quadric function of the (x) whose coefficients (A,^) are the 

 minors of the determinant formed by the coefficients (a.^) of T, (dis- 

 criminant of T,) and for (/3) 



Z'2-?'3 Z'a-^i ii-h' ^^-^^ ^-^^ +2/3) -const. 



The case where the last condition is satisfied has been investigated by- 

 Weber ' when the impulse of the motion reduces to a single force — in other 

 words, when the state of motion can be produced by a single blow 

 applied to a point rigidly connected with the body. Out of the sixteen 

 Theta functions of the first order of two variables, it is possible, in several 

 ways, to choose nine, whose ratios to a tenth, multiplied by proper con- 

 stants, are suitable to express the direction cosines of one set of rectan- 

 gular axes to another. "Weber shows that, taking the two variables to be 

 linear functions of the time a t + (d, a^ t +/3i, and those nine ratios to 

 represent the direction cosines of the axes fixed in the body, to the axes 

 fixed in the space, it is possible, if Clebsch's last condition above is satis- 

 fied, to determine the constants, so that three other ratios may represent 

 the rotations about the set of axes fixed in the body, and the remaining 

 three the. rotations about the axes fixed in space, provided the motion is 

 such that there is no impulsive couple. There remain four constants to 

 be determined by the initial conditions, viz., two relations between three 

 moduli of the Theta functions and the two constants /3 /3p Four cases 

 occur as in the previous investigations, which depend on the initial state. 

 These are distinguished in the latter part of the paper. He also treats 

 the equations in another way by direct integration by means of hyper- 

 elliptic integrals. 



The motions of a solid about a fixed point in fluid under the action of 

 no forces, and about a fixed axis under the action of gravity, have been 

 investigated by Michaelis.^ 



In a remarkably suggestive paper in the ' Proceedings of the London 

 Mathematical Society,'^ Lamb has applied Ball's theory of screws to the 

 question of the steady motion of any solid in a fluid. It is easy to see 

 that there are a simply-infinite system of steady motions, each being a screw 

 motion, whose axis lies on a certain skew surface. The axis of each 

 screw must coincide with the axis of the generating wrench, but their 

 pitches are not necessarily the same. If the ratio of the impulsive force 

 to the rotation set up is given, there are three screws of steady motion 

 perpendicular to each other, though not necessarily intersecting. Amongst 

 the infinite system of permanent screws, it is possible to choose sets of 

 six mutually conjugate screws amongst which there is one set which 

 contains three of infinite pitch (which correspond to the three steady 

 translations), and three others which are such that the necessary gene- 

 rating wrenches have zero pitch, i.e. reduce to impulsive couples. This 

 latter set of three is important, as by its means it is possible to construct 

 the motion when the generating wrench reduces to any couple whatever. 

 The following is Lamb's method of representing the motion. The three 



' 'Anwendung der Thetafvmctionen zweier Veranderlicher auf die Thcorie der 

 Bewegung- eines festen Korpers in einer Fliissigkeit,' Math. Aimalcn, xiv. p. 17.^. 



= ' Mouvement d'un solide dans un liquide,' ArcMves Nierlandaisex des Seiences 

 exactes ct natiirelles, Harlem, viii. 



2 ' On the motion of a solid through an infinite m^s of liquid,' Proe. Lond. Math, 

 Soc, viii, p. 273 (1877\ 



