654 MOORE. 



the word perpendicular shall be used to mean complete perpendicu- 

 larity. 

 The formular for the reduction of the various products of 1- and 2- 



vectors are, 



(1). a-{h^c) = {a-c)h — (a-b)c 



(2). ia^b)-C = a-ib-C) = -b-(a-C) 



(3). ax(6-C) =(axC)-b- {a-b)C 



(4). {b-C)-A= -b{C-A)-^C-{b^A) 



where a, b, c are l-vectors and A, C are 2-vectors. 



Now having these definitions an infinitesimal rotation ^ parallel to 

 a fixed plane Mi is defined by the equation 



r' = r -\- Mvrdi. 



The length of r' is equal to the length of r. For 



r'.r' = r-r + 2r- {Mi-r)dt = r-r 



The product r • (Mi • r) vanishes since Mi • r is defined as a vector in 

 Ml perpendicular to r. A general rotation can be considered as made 

 up of rotations parallel to a number of independent planes. The 

 equation for such a rotation is 



r'=r + (Ml + il/2+ .... -\-Mk) • r dt. 



The sum of k, simple plane vectors is a complex ^ plane vector or 

 2-vector. Therefore we may write the rotation in the form 



(5) r'=r-\-M-rdt 



where M is a complex 2-vector, that is, is not equivalent to a simple 

 2-vector. The canonical form then which (5) may take depends on 

 the form in which 31 may be written. Then we shall first show that 

 M can always be resolved into the sum of p mutually perpendicular 

 planes if we are working in a space of 2 p dimensions. We first 

 consider the cases of four dimensions and five dimensions and then 

 generalize to space of any number of dimensions. 



6 A rotation is defined as a rigid motion leaving one point fixed. 



7 Plane vectors in 4-space, for example, are analogous to lines in 3-space. 

 We know that the sum of two lines is not a line unless the lines intersect. 

 The same is true of plane vectors, their sum is a complex or complex vector 

 unless the two simple plane vectors have a line in common. We shall use 

 complex as equivalent to complex vector. 



