WILSON AND LKVVIS. — RELATIVITY. 505 



The preceding ideas can he used to give new definitions of the inner 

 and outer prochicts of two vectors. Namely, 



a«b = distance of a times distance of b times cosh Z (a, b), 

 axb = distance of a times distance of b times sinh Z (a, b), 



it being understood that the latter quantity is not a scalar but a pseudo- 

 scalar. If a and b are vectors of the same class the angles are real, 

 and the equations are essentially identical with those which have been 

 previously derived. If a and b are (5)-vectors the distances are purely 

 imaginary and the product a'b is negative if the vectors issue into the 

 same "quadrant." If a and b arc of dilfcrent classes, and the angle 

 between them complex, we may use in place of these complex angles 

 their complementary real angles by the aid of the familiar formulas 



cosh (0 + §7ri) = isinh0, sinh ((^ + hT^i) = icosh(f>. 



Mass.\chusetts Institute op Technology, 

 Boston, Mass., May, 1912. 



TABLE OF NOTATIONS. 



General Symbols. 



1-yectors, lower case Clarendons, a, b, c,...; 



their magnitudes, corresponding Italic, a, b, c,...; 



their components (algebraic magnitudes), ai, 02, a^, 04, etc.; 



their (vector) space components, as, bs, Cs,.... 



ki. kj, ks, unit coordinate space vectors; 



k4, unit coordinate time vector. 

 2-vectors, Clarendon capitals, A, B, C,...; 



their magnitudes, corresponding Italic, A, B, C,...; 



their comjKjnent.s, Au, Aoz,..., A34', etc.; 



ki:, kjs,..., k34, unit coordinate 2-vcctors. 

 3-vector.s, 1\i(lor black capitals, B, JB, C,...; 



their magnitudes, corresponding German, 3(, 58, 6,...; 



their components, A234,..., A123; 



k23j...., ki23, unit coordinate 3-vectors (the last, "space"), 

 unit pseudo-scalar, ki234. 

 sign of the outer product, small cress, x. 

 .sign of the inner product, heavy dot, •. 

 sign of the complement, asterisk, a*, A*,.... 

 three-dimen.sional differentiating operator, del, V. 

 four-dimen.sional differentiating operator, quad, O- 

 dyadics, Greek capitals, *,... (idemfactor, 1). 



