TBANSACTIONS OF SECTION A. 535 



and consequently 



d.V, rf.fj _ ^2 , 



where .r^, .rj are functions of x^, x^ respectiyely : hence taking the logarithm and 

 differentiating successively with regard to >rj and .r^ we have 



A A \os (^-^^^^Sz\^o 

 dx-i dx^ \dx^ ' dx^' ' 



which is the required partial differential equation of the third order. 



This differential equation has a simple geometrical signification. Consider 

 three consecutive positions of the line meeting the cubic curve in the points 1, 2, 3 ; 

 1', 2', 3' ; V, 2", 3" respectively : qua equation of the third order, the equation 

 should in effect determine 3" by means of the other points. And, in fact, the three 

 positions of the line constitute a cubic curve ; the nine points are thus the inter- 

 sections of two cubic curves, or, say, they are an ' ennead ' of points ; and any eight 

 of the points thus determine uniquely the ninth point. 



10. On the Differential Equations satisfied hy the Modular Equations. 

 By Professor H. J. S, Smith, M.A., F.Ii.8. 



11. On the q-Series in Elliptic Fu/nctions, 

 By J. W. L. Glaisher, M.A., F.B.8. 



12. On the Elucidation of a Question in Kinematics hy the aid of Non- 

 Euclidian Space. By Eobekt S. Ball, LL.D., F.B.8. 

 It is well known that the family of quadric surfaces denoted by the equation — 

 (a-k) x"+(b-k) if + (c~k) z^ + (a-k) (b-k) (e-k) = 



denotes the screws about which a body with freedom of the thii'd order can twist. 

 The parameter k is the pitch. 



It is easily shown that this family of surfaces is inscribed in a common tetra- 

 hedi'on of which the faces are the imaginary planes denoted by — 



\/b — cx+ /v/e — ai/+ ^a — bz+ A/b — c \/c — a t^a — b = 



and the three other expressions which can be produced by the indeterminateness of 

 the signs of the radicals. 



Each of these surfaces also passes through the four points in which the two 

 cones — 



x^ + 3/^ + s^ + 

 ax^ + by^ + cs* = 

 are cut by the plane at infinity. These points lie one by one on each of the four 

 faces of the tetrahedron. 



It occurred to me that these properties of the system of sui"faces were probably 

 only the ' sui-vivals ' of a more interesting geometrical system in non-euclidian space. 

 These anticipations have been fulfilled, and the result is to give a complete geo- 

 metrical theory of the statics and kinematics of a rigid body with three degrees of 

 freedom in non-euclidian space. 



The most general motion of a body is produced by rotations around a pair of 

 lines which are conjugate polars with regard to the absolute. For convenience I 

 denote these rotations by a> cos a and <o sin a. The total amplitude of the motion 

 is (o, while its_ pitch is denoted by a. A pair of conjugate polars with their asso- 

 ciated a form in non-euclidian space the analogous conception to a screw in ordinary 

 space. 



The first step in the theory is to find the condition that a twist about one screw 

 shall do no work against a wrench on another screw. 



Let A and A' be the conjugate polars forming the first screw whose pitch is a, 



