138 Dr. L. Silberstein : Further Contributions 



constructed *. We shall henceforth denote any such triad 

 of normal unit vectors by i, j, k, assuming this to be their 

 order in a right-handed system. 



It may be interesting to notice that the end-point P of 

 the vector 



P=i+jfk (16«) 



is the cross of the three tangent planes to Q at the end- 

 points i,j s k of these normal vectors. More generally, if 

 a, b, c are any three unit vectors " equally inclined " to one 

 another, i. e. such that ab=bc = ca = /V, the planes touching 

 Q at a, h, c cross in the end-point of the vector 



P=^-. ...... (16). 



The proof, similar to that of equation (2), offers no 

 difficulty. 



All other details concerning the passage from a plane to 

 three-space may be left to the care of the reader. He will 

 easily convince himself that the distributive property, 



A(B + C)=AB + AC, 



with its consequences, will remain valid, whether A, B, C are 

 (or can be made) coplanar or not. 



One of the consequences will be the generalized Pytha- 

 gorean theorem for three dimensions, 



i7 2 = D 2 = A 2 + B 2 + C 2 , 



where A, B, C are any normal vectors and D their sum. Thus 

 also the line-element, i. e. the tensor ds of the vector 



ch = i dx -\-3dy-\-kd~, 

 will be given by 



ds 2 =dx 2 + df + dz 2 , (17). 



the extension of (14) to space. The introduction of polar 

 coordinates and allied questions do not call for any 

 explanations. 



7. Vector Product of two Vectors. — Once in possession of 

 the concept of orthogonality we can define the vector 

 product of two vectors in much the same way as in common 

 A^ector algebra. 



Let A = ^4a and B = i?b, and let he the concave angle 



* In technical language, OT a T h T c will be a self-polar tetrahedron, 

 with respect to k. 



