468 PROCEEDINGS OF THE AMERICAN ACADEMY. 



use another construction which is essentially identical with this. 

 Let dr be any vector drawn from one boundary line to the other. 

 Then (rfrxf )*f // is the same vector as the one just obtained. Although 

 the method of obtaining this vector may seem somewhat artificial, 

 the vector is, however, a definite vector obtainable from the field 

 without any choice of axes. 



46. These methods fail completely when the vector field is com- 

 posed of singular vectors. Let us consider instead of a material rod, 



a segment of a uniform ray of light. If this 

 can be represented by a continuous vector 

 field bounded by two lines representing the 

 loci of the termini of the segment then all 

 these vectors must be singular. Let 1 be 

 (Figure 25) the value of the vector through- 

 out the field. It is evident that we cannot, 

 / ^^ as in the former case, draw any line across 



Figure 25. the field perpendicular to 1. The second 



method likewise fails because it would involve 

 the magnitude of 1 which is zero. Moreover it can be stated that 

 there is no method whatever, independent of any choice of axes, 

 which will enable us to change from this continuous distribution of 

 the light to a set of light particles. Conversely it is equally true that 

 given a system of light particles moving in a single ray it is quite 

 impossible to replace them by means of any continuous distribution, 

 and this is true no matter how small and numerous and close to- 

 gether these particles are. This statement regarding singular vectors 

 will be seen to hold also in space of higher dimensions,*^ and is of 

 fundamental importance. 



While it is impossible, therefore, to find continuous and discontinu- 

 ous distributions of singular vectors which are ec^uivalent to one 

 another, it is possible to obtain by four dimensional methods out of 

 a specified region of a singular vector field a single \ector or group of 

 discrete vectors uniquely determined by that vector field but quadratic 

 instead of linear in the vectors of the field. Consider any portion of 

 the field bounded by two singular vectors sufficiently near together. 

 Let 1 be the vector of the field, and then if dr is any vector drawn from 



41 In the ca.se of the pecuhar geometry of a singular plane (§31), the interval 

 dr from one singular lino to another is independent of the direction of dr. It 

 i.s therefore possible to replace the field 1 between two boundarj' lines by the 

 single vector Idr linear in 1. Thus there are exceptional singular fields in 

 higher dimensions for which the passage from continuous to discrete and vice 

 versa may be accomplished. 



