626 Mr. H. Bateman on the Relation 



In order to specify a space-time vector o£ the first kind 

 through a given sphere of observation A when the above 

 representation is used, it is sufficient to know the position of 

 the centre of similitude of the spheres A, B (attention being 

 paid to the signs of their radii in determiuing the choice of 

 one out of the two centres of similitude), and a number 

 indicating the magnitude of the vector *. If the centre of 

 similitude lies within the sphere of observation, the vector is 

 said to be temporal, if it lies outside, the vector is said to be 

 spacial f. Two vectors are said to be orthogonal J when the 

 corresponding centres of similitude are conjugate points with 

 regard to the sphere of observation. A vector orthogonal to 

 a temporal vector is necessarily spacial, but the converse does 

 not hold. 



The relations which two spheres B and C (or a space-time 

 vector of the first kind not passing through A) bear to the 

 sphere of observation determine a species of space-time vector 

 of the second kind containing A. A space-time vector of 

 the second kind is specified by six components 



(E„ E y) B„ H„ H y) H.), 



and the special type at present under consideration is 

 characterized by the existence of the relation 



A more general space-time vector of the second kind may 

 be obtained by adding the components of two special space- 

 time vectors of the second kind. 



A special space -time vector of the second kind may be 

 specified by means of a line, viz. the axis of similitude of the 

 spheres A, B, C and a number to indicate the magnitude of 

 the vector. The magnitude of the vector may be taken to be 

 equal to this number multiplied by the area of the triangle 

 PQR, where F, Q, R are the points of contact of a common 

 tangent plane of the three spheres A, B, C. 



* The magnitude of the vector may be taken to be equal to this 

 number multiplied by the length of a common tangent of the two 

 spheres. The magnitude of a vector may vanish although its compo- 

 nents do not. 



"j" These terms were introduced by Minkowski, but are defined analyti- 

 cally. If (X, Y, Z, cT) are the four components of a space-time vector 

 of the first kind, it is said to be temporal or spacial according as 



c 2 T 2 5X 2 +Y*+Z 2 . 



+ Minkowski uses the word normal. Two vectors whose components 

 are (X, Y, Z, cT) (X x , Y x , Z l? cT x ) are normal to one another if 



XX^YY.+ZZ^TT,. 



