326 



TRANSACTIONS OF SECTION A. 



(2) If a rational function is adjoint for the value z =Oothe degree of its re- 

 duced form is <N — 1. 



(3) The reduced form of a rational function, adjoint for the value r = a, is 

 integral with regard to the element r — a if the equation (1) is integral with 

 regard to this element. 



2. A Theorem connected with Six Lines in Space. By H. Bateman, M.A. 



Let PP', QQ', PuR' be three pairs of non-intersecting straight lines. The 

 locus of a point 0, which is such that the three chords from to the pairs of 

 lines lie in a plane, is a quartic surface which contains the six lines PP'QQ'RR'. 



If LL', MM', NN' are the common transversals of QQ'RR', RR'PP', 

 PP'QQ', the quartic surface derived from LL', MM', NN' is the same as that 

 derived from PP', QQ', RR'. 



If QQ'RR' are generators of one system of a hyperboloid, there will be an 

 infinite number of transversals of type LL', and these will all lie on the hyper- 

 boloid. The hyperboloid must form part of the quartic surface, for in the 

 general case the lines LL', MM', NN' all lie on the quartic. The complete locus 

 now consists of two hyperboloids, the one just mentioned and a second one, 

 which must contain the lines MM', NN', PP'. 



We thus have the theorem. If QQ'RR' are non-intersecting generators of a 

 hyperboloid and PP' two arbitrary lines, the lines MM'NN' are generators of a 

 hyperboloid. 



3. The Canonical Form of an Orthogonal Substitution. 

 By Harold Hilton. 



Suppose (X v X 2 , • • •, X m ) is a pole of a real orthogonal substitution A on the 

 variables x v x. if . . ., x m which corresponds to a root A of the characteiistic equa- 

 tion of A ; where A. 2 ^ 1. The inverse of the real orthogonal substitution B with 

 matrix 



b n b u 

 b n o 22 





.6. 



where b n = \ (X,+ X t ), i b n = I (Z,-X t ), (*— 1, 2, . . ., m), 



will transform A into a real orthogonal substitution, which splits up into a real 



for the matrix of B, where Xi 2 + X.^ + . . . + X*„=1, and obtain a similar result. 



A similar process holds good for an unreal orthogonal substitution A, if and 

 only if A has a simple invariant-factor. 



If A is real and orthogonal, the process can be repeated as often as desired, and 

 we thus obtain a practical method of transforming A into its well-known canonical 

 form by means of a real orthogonal substitution. 



