LEWIS. — ON FOUR-DIMENSIONAL VECTOR ANALYSIS. 1G7 



be represented by a small letter in heavy tyiie (e g., a, s). A 2 -vector 

 will be represented by a capital letter in heavy type (e. g., A, S). 



Let us consider a coordinate system of three perpendicular axes, 

 i?'i, a'2, a^a, and represent by ki, kg, kg, the three unit vectors in these 

 three directions. If the lengths of the components of a 1 -vector a on 

 the three axes are cii, a^, as, then 



a = ttiki -f ajki + a^iis- (1) 



In general the addition and subtraction of 1 -vectors follow the law.® 



a ± b = (f/i ± b,) ki + (rt2 ± ^^2) ko + («3 ± bs) K (2) 



Similarly we may project a surface vector, or 2-vector, upon the three co- 

 ordinate planes determined by a\, X2. ; ^1, ccz ; x^, x^. The unit 2-vectors 

 in these planes we will denote by ki2, kjs, kjsJ and the areas of the 

 projections of a 2-vector A by A^ ,^13, A^^. Then, 



A = A 12^12 + Aukis + A^zk-os (3) 



A±B = (Au ± ^12)^12 + (^13 ± ^13)^-13 + (^23 ± ^23)^-28. (4) 



Further we may adopt the convention, 



ki2 = — k2i ; ki3 = ksi ; K23 =^ k32 (5) 



which requires the further convention, 



A12 = — A21 ; Ai3 = — Azi ; A23 = — ^32- (6) 



Just as ki represents a vector of unit length, ki2 one of unit area, 

 ki28 will represent unit volume. It is the unit 3-vector or, in three- 

 dimensional space, the unit pseudo-scalar. We shall adopt the 

 convention 



kl23 ^ k3i2 ^ k2si = •'^132 ^= — •^213 ^= •'^321. (7_) 



Equations (5) and (7) may be expressed in the following general 

 rule which we shall also adopt in space of higher dimensions : Inter- 

 changing any two adjacent subscripts of a unit vector changes the sign 

 of the vector. 



' The addition both of 1-vectors and 2-vectors is best defined geometri- 

 cally. (See Grassmann, Ausdehnungslehre von 1844, p. 78.) The introduction 

 of coordinate axes brings a foreign element into pure vector analysis, but on 

 the other hand it will permit us to translate the vector equations more readily 

 into Cartesian equations. 



'In this case we depart from our rule that 2-vectors shall be represented 

 by capital letters. On account of the subscripts there will be no confusion. 



