440 PROCEEDINGS OF THE AMERICAN ACADEMY. 



may write b = b' + b", Avhere b' is in the space of a and C and b'^ 

 is perpendicular to a and C. Then by (36) 



ax(b'.C) = (axC)-b' — (a.b')C, 



and ax(b".C) = (axC).b" — (a.b")C 



holds identically, since each of its terms vanishes. Hence by addition 

 (36) is seen also to hold in general. 



Some products involving more than three 1-vectors are of frequent 

 Occurrence. By (35) and (34) we may write immediately 



(axb).(cxd) = (a.c)(b.d)-(b.c)(a-d) = ^'^ f'*^i. (39) 



ib«c b«d| 



In a similar manner we may prove 



a*d a*e a^f 



(axbxc) • (dxexf ) = b«d b«e b'f 



c«d c«e C'f 



and 



(axb) • (cxdxe) = (axb) • (dxe) c + (axb) • (exc) d + (axb) • (cxd) e. 



These formulas are all valid in space of any dimensions. 



The Biffcrentiatincj Operator V. 



34. In discussing the differential calculus of scalar and vector 

 functions of position in space, the vector differentiating operator V is 

 fundamental. The definition of this operator may be most simply 

 obtained as follows. Consider a scalar function F of position in space. 

 Let dx denote any infinitesimal vector change of position, and let dF 

 denote the corresponding differential change in F. Then let V be 

 defined by the equation 



dF = dX'\/F. 



Now VF is a vector. If dv is a vector in the tangent plane to the 

 surface F = const., dF is 0, and as dvX/F then vanishes, the vector 

 dr and \/F are perpendicular. Hence V/'^ is a vector perpendicular 

 to the surface F = const. Now V F may be a vector of the (5)- 

 class or of the (7) -class, and the tangent plane is then respectively 

 a (7)-plane or a (5)-plane.^^ 



29 In our non-Euclidean geometry vF will not be a vector in the line of the 

 greatest change of F. If dr be written as U ds, where U is a unit vector in the 



