472 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



(1 —V COS (j>) ds/ Vl — v^, where ^ is the angle between dq and AD. 

 Hence 



dg 1 — V cos(j) . , 



y- = /4 , Ids. 



dr Vl 



^2 



(89) 



We have gone through this somewhat compHcated calculation for 

 the three dimensional case because of the greater ease of visualisation 



Figure 27. 



and because the results obtained are applicable without essential 

 change to four dimensions. Again let ds be a segment of any (5)- 

 curve each point of which determines a forward hypercone. Let us 

 consider the four dimensional vector field 1 bounded by the two 

 limiting forward hypercones, 1 at every point lying along an element 

 of one of the hypercones whose apex is on ds. Any (7)-planoid will 

 intersect the limited vector field in a three dimensional volume bounded 

 by the intersections of the two limiting hypercones with the planoid; 

 these surfaces of intersection appear in the planoid as two nearly 

 concentric spherical surfaces. 



If as before the vector field is divided into infinitesimal portions, so 

 that the volume of intersection is divided into the infinitesimal vol- 

 umes f/S, each of which is approximately a rectangular parallelepiped, 

 and one of the surfaces of intersection is thus divided into the infi- 

 nitesimal portions dS such that dqxdS = c/S, then for each infinitesimal 

 portion of the field we may at any point obtain as above the vector 

 dg = (rfS^l)*l. Then precisely as in the previous case *^ 



42 In the peculiar three dimensional geometry of a tangent (singular) planoid 

 there is one; set of parallel singular lines, and every plane in the planoid is 



f)erpendicular to these lines. Every cross-section of a given tube of singular 

 ines has the same area. 



