500 PROCEEDINGS OF- THE AMERICAN ACADEMY. 



The transformation r' = r* A, where A is a uniplanar 2-vector, can 

 be regarded geometrically as an annihilation of that part of r which 

 is perpendicular to A, and a replacing of the component of r in A by a 

 perpendicular \Tctor magnified in the ratio of ^ to 1. The transforma- 

 tion r' = (r • A) • A therefore annihilates components perpendicular to A, 

 and reverses components in A, multiplying them further by A 'A. 

 Hence if A is a (7)-plane, the transformation in that plane is rotation 

 through a straight angle combined with a stretch as ^" : 1 ; whereas 

 if A is a (5)-plane, the transformation is one of stretching only, as 

 A* A is negative. 



In case A is biplanar we may resolve it into its two completely 

 perpendicular parts, A = B + C, where B is a (7)-vector and C 

 a (6)-vector. Then the equation 



r' = (r.A).A = (r.B).B + (r.C).C 



holds by virtue of the fact that r*B is perpendicular to C, and r-C 

 perpendicular to B. Hence the transformation r' = (r'A)'A consists 

 of rotation through a straight angle and stretching in the ratio B'A 

 for components along B, and of stretching alone in the ratio C^:l 

 for components along C. 



The transformation r' = (r'A)«A + (r'A*)'A* is now readily seen 

 to be a stretching of components along B or C in the ratio {B~ + C~) : 1 

 combined with a reversal of the direction of the components along B. 

 If this transformation were repeated, the result would be to stretch 

 all vectors in space in the ratio (B- + C~)~: 1 . But 



(^2 _|_ ^2)2 = (^2 _ (72)2 4. 452(72 = (A.A)2 + (A.A*)^. 



Hence the square of i (^'^ + ^*-^*) is i [(A- A)^ + (A-A*)-] I, 

 a multiple of the idemfactor. This is the geometric interpretation 

 of a result obtained analytically by Minkowski. 



63. From the definition (48) of the differentiating operator <3>, 



it follows that the expression <0f, where f is a 1-vector, is a dyadic. 

 This definition may frec^uently be applied directly and with ease to 

 determining the dyadic <^t, and renders unnecessary the expansion 

 of <^f in terms of its components. For if the value of di for four 

 independent displacements dr can be found, the dyadic is thereby 

 completely determined, and in some cases can immediately be written 

 down by inspection. This was the method pursued in § 44. The 

 dyadic itself, however, was not then desired except for the purpose 



