LEWIS. — ON FOUR-DIMENSIONAL VECTOR ANALYSIS. 173 



Here as elsewhere our equations differ from those in coiinuon use 

 whenever the product of a 1-vector and a 2-vector is concerned. 



The Vector Analysis of Four Dimensions. 



The revised system of three-dimensional vector analysis has been 

 elaborated somewhat fully in the preceding section, since the methods 

 there adopted may be used without any modification in developing the 

 vector analysis of space of higher dimensions. 



Let us consider a four-dimensional space in which any two points 

 uniquely determine a straight line, any three points not in a line 

 uniquely determine a plane, and any four points not in a plane uniquely 

 determine a straight or Euclidean 3-space. This may be called a 

 Euclidean four-dimensional space. 



In such a space let us construct four mutually perpendicular co- 

 ordinate axes, .ri, x^, .r^, a'i. The 1 -vectors of unit length in these four 

 directions we may call kj, kj, kg, k^. Each pair of axes determines a 

 plane, thus forming six coordinate planes. The 2-vectors of unit area 

 parallel to these planes we may call k^a, kig, ki4, kjs, kai, k34. These six 

 planes are mutually perpendicular. Moreover the plane ki2 is coni- 

 pletely perpendicular to the plane kgi, in the sense that every line in 

 ki2 is perpendicular to every line in kj^. The same is true of the pairs 

 k23, ki4, and ki3, ko*- 



Each set of three axes determines a straight 3-space and the four 

 coordinate 3-spaces thus determined may be represented by the unit 

 3-vectors kias, ki24, ki34, k234. Finally all four axes together determine 

 the unit 4-vector or pseudo-scalar, ki234. 



A 1-vector may be represented as the sum of its projections on the 

 four axes, 



a = rtiki -f a.^s.. -f a3k3 + <:?4k4. (35) 



A 2-vector may be represented as the sum of its projections on the six 

 coordinate planes. 



A = -4i2ki2 + ^liskis + -4i4ki4 + ^423k23 -\- ^l24k24 + ^34k34. (36) 



A 3-vector may likewise be represented as the sum of its four projec- 

 tions on the coordinate 3-spaces.^^ 



The addition and subtraction of vectors follow the same rules as 

 in the case of three dimensions (equations 2 and 4). Moreover both 



" We shall not use the 3-vectors often enough in this paper to justify the 

 introduction of a special symbol for them. 



