470 PROCEEDINGS OF THE AMERICAN ACADEMY. 



section of a perpendicular planoid is a single vector (or the sum of a 

 group of vectors) which may replace the original field within the tube. 

 Or if f represents the vector field and f/S the 3-vector cut off on any 

 planoid by the tube, then the same result as before ina^^ be obtained 

 by the operation (f/Sxf)*f //. 



In the case of singular vectors we encounter the same difficulties 

 as in two dimensions. Let us consider a field of singular 1 -vectors 1, 

 and a portion of this field cut off by a small tube of lines parallel to 1. 

 A little consideration shows that it is impossible by any means what- 

 ever to replace this portion of the field by a single equivalent vector 

 along 1. It is possible, however, as before to obtain a single vector 

 quadratic in 1 and determined by the given portion of the field. Let 

 c^S be the 3-vector volume cut off on any planoid by the tube. Then 

 (f/Sxl) is independent of the planoid chosen, and (f/Sxl)*l= dg is 

 the vector thus determined. 



47. Now it is impossible to distribute the vector just obtained 

 over that portion of the four dimensional spread which has gi\en rise 

 to it. But there is, nevertheless, in one case another kind of dis- 

 tribution which is possible and which possesses considerable interest. 

 In order to introduce the somewhat difficult construction which is 

 necessary in this case let us investigate first a particular type of 

 singular vector field in three dimensions. Let c/s be a small vector 

 segment of a (5)-curve. Each point of this segment determines a 

 forward cone. The field which we wish to consider is such that at 

 each point the vector 1 is along an element of the cone and of any 

 interval which is a continuous function of position. This construc- 

 tion gives a limited field bounded by the two forward cones from the 

 termini of the segment ds. Let a plane cut across the two cones. 

 The region of this plane intercepted between the two boundary cones 

 is the surface lying between two nearh^ concentric circles Let dS 

 be an element of this surface. Now just as before the vector 

 (rfSxl)*l = dg may be formed and is dift'erent for each element c/S. The 

 singular lines drawn from all the points bounding dS to tlie corre- 

 sponding points of the segment ds determine a sort of tube of nearly 

 parallel singular lines. The value of dg for each tube is at each point 

 independent of the particular position of the plane through that point 

 whose intersection with the tube is f/S. If therefore the whole field 

 is divided up into an infinite number of such tubes, the infinitesimal 

 vectors of the second order in 1 obtained for the several tubes are 

 at each point independent of the plane which was used in constructing 

 them. 



