652 MOORE. 



The only papers that I know of bearing on rotations in hyperspace 

 are given below: ^ 



1. Introduction. In terms of the Gibbs Vector analysis an 

 infinitesimal rotation in three dimensions can be expressed by the 

 formula ^ 



/ = r + a^rcJt, 



where a is a vector along the axis of the rotation and r is any vector 

 through the fixed origin. If the rotation is considered as parallel to 

 a fixed plane determined by the vectors b, c then it can be represented 

 by the formula 



r' = r -\- {bxc)xr dt. 



By the Gibbs definition of the cross product (a vector perpendicular 

 to the plane of the two vectors and of magnitude equal to the product 

 of the lengths of the two vectors into the sine of the angle between 

 them, so arranged that the three vectors b, c, b >^ c for a right handed 

 system) this last expression is equivalent to the first. If we wish to 

 extend this to higher dimensions we cannot have a form equivalent 

 to the first since the cross product of two 1 -vectors cannot be con- 

 sidered as a 1-vector as, in that case, two l-vectors do not uniquely 

 determine a 1 -vector. We will then have to start with a new set of 

 definitions of the products. It must be kept in mind that we may have 

 vectors of different dimensions as l-vectors, 2-vectors, 3-vectors, etc., 

 of one, two, three, etc., dimensions. I shall here use the vector 

 analysis already set up by Wilson and Lewis.* 



The cross product of two-vectors extending from the same origin is 

 defined as the parallelogram defined by the two vectors. The product 

 is then a 2-vector. The magnitude of the product is equal to the area 

 of the parallelogram. Similar definitions are given for the cross 



2 F. N. Cole: On rotations in space of four dimensions. American Journal, 

 12, 1889, page 191. 



P. H. Schoute: Le deplaceijient le plus general dans I'espace a n dimensions 

 Annales de I'Ecole Polytechnique de Delft, 7, 1891. 



Bemporad: sui grupiii dei movimenti. Annali della r. scuola normale sup. 

 di Pisa, 8, 1904. 



E. E. Levi: Sui gruppi di movimenti. Atti dei Lincei Series 5, 14, part 1, 

 1905. 



3 Gibbs-Wilson Vector Analysis, page 99. 



4 Space-time manifold of relativity. The non-euclidean geometry of 

 mechanics and electromagnetics. These Proceedings, 48, number 11, 1912. 



