90 VECTOR ANALYSIS. 



lie in any same real plane.) If ci.bxc is not equal to zero, the 

 equation shows that any fourth bivector may be expressed as a sum 

 of a, b, and c with biscalar coefficients, and indicates how these 

 coefficients may be determined. 

 11. The equation 



(r.ci)bxc+(r.b)cxa+(r.c)axb==(axb.c)r 

 is also identical, as may easily be verified. If we set 



c = axb, 

 and suppose that r.a = r.b = 



the equation becomes 



(r.axb)axb = (axb.axb)r. 



This shows that if a bivector r is perpendicular to two bivectors 

 Q and b, which are not parallel, t will be parallel to axb. Therefore 

 all bivectors which are perpendicular to two given bivectors are 

 parallel to each other, unless the given two are parallel. 



[Note by Editors. The notation |* | $c~ l > used on page 64, was later improved by the 

 author by the introduction of his Double Multiplication, according to which the above 

 expression is represented by 4> 2 , and |<t>[ by 4> 3 . See this volume, pages 112, 160, 

 and 181. For an extended treatment of Professor Gibbs's researches on Double Multi- 

 plication in their application to Vector Analysis see pp. 306-321, and 333 of "Vector 

 Analysis," by E. B. Wilson, Chas. Scribner's Sons, New York, 1901.] 



