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VII. 



THE GEOMETEICAL MEANING OF CAYLEY'S FORMULiE 

 OF ORTHOGONAL TRANSFORMATION. 



By C. H. HINTON, Patent Office, U.S.A. 



[communicated by PKOPESSOE a. C. HADDON, F.E.S.] 



Read November 29, 1902. 



Cayley has given the geometrical significance of his formulge of 

 transformation in the case of three axes. In Vol. xn., American 

 Journal of Mathematics^ Cole has shown the geometrical significance 

 of the formulae in a special case for four dimensions. In order to 

 discuss the question of the significance of the general case in four 

 dimensions, I will write down all the forms which can come from 

 any combination of axes and angles. It will be seen that Cayley's 

 forms in the general case are incapable of geometrical significance in 

 terms of axes and angles. 



A very convenient form of quaternion symbolism can be used of 

 space of 2" dimensions, which I will adopt in the present discussion. 



Let ii y, ^, h be unit vectors mutually perpendicular. Assume 



4-2 =y2 = ^2 = ^2 = + 1. 



Let a transposition be accompanied by a change of sign. Taking the 

 two possible Hamiltonian circuitings 



ijlch = + 1 , ijkli = - 1 , 



let us distinguish between the two systems of equations derivable by 

 calling those derived from ijhh, = + 1 the A kind, and those from 

 ij'kh = - 1 the B kind. 



To distinguish when the latter equation is used, introduce a symbol 

 w, which has simply the significance that the vector symbols after it 

 are combined— according to the ijlcli = - 1 laws. 



From the A form 

 we get ijh = A, 



from the B form 

 we get ijh = - h, 



Hence, introducing the syn 



R.I. A. PKOC, VOL. XXIV., SEC. A.] U 



ijkh = + 1, 





ij = hk, 



ji = kh ; 



ijkh = - 1, 





ij = - hk, 



ji = hk. 



ol to, we get oy' 



i = uihk, &c. 



