452 PKOCEEDINGS OF THE AMERICAN ACADEMY. 



The complement of a singular 1 -vector is a 3-vector which itself 

 is evidently singular. It is the planoid tangent to the hypercone 

 through the given element. ^^ It contains, besides the pencil of singu- 

 lar planes through the element of tangency, only (7)-planes. 



We may take this opportunity of summarizing the properties of 

 singular vectors in general. The inner product of any singular vector 

 by itself is 0. Every singular vector is perpendicular to itself and to 

 every singular vector lying within it. The magnitude of a singular 

 vector is zero. This does not imply that such a vector is not a 

 definite geometric object, but only that the interval of a singular 

 1-vector, the area of a singular 2-vector, and the volume of a singular 

 3-vector are zero when compared with non-singular intervals, areas, 

 and volumes. 



The visualization of the geometrical properties of a four dimensional 

 and especially of a non-Euclidean four dimensional geometry is 

 extremely difficult. It is of course possible to rely wholly on the 

 analytic relations, and thus avoid the difficulty. But we believe that 

 it is of the greatest importance to realize that we are dealing with 

 perfectly definite geometrical objects which are independent of any 

 arbitrary axes of reference, and that it is therefore advisable to make 

 every effort toward the visualization. It seems probable that Min- 

 kowski, although he employed chiefly the analytical point of view 

 in his great memoir, must himself have largely employed the geo- 

 metrical method in his thinking. 



The Differentiating Operator <0>. 



40. By analogy we may in four dimensions define the operator <(^, 

 called quad, by the equation 



r/( )= dr-Oi ). (48) 



When referred to a set of perpendicular axes, quad takes the form 



0= k,^ + ko^ +k3|--k4;.-' (49) 



d.i\ dx-z axs dxji 



and like V it may be regarded formally as a 1-vector. 



33 The gecinetry in a singular planoid is analogous to that in a singular plane 

 (§31). In this 3-space there are two classes of lines, singular lines, all of which 

 are parallel to each other, and non-singular hnes, (7)-lines, all of which are 

 perpendicular to tlie singular lines. Similarly there are two classes of planes, 

 singular planes, all of which are parallel to the singular lines, and non-singular 

 (7)-planes, which are perpendicular to every singular plane. Volumes are 

 comparable with one another but are all of zero magnitude as compared with 

 a volume in any non-singular planoid. 



