1882.] Mr H. Cox, On the application of Quaternions. 195 



l* =-jJ 2 + (£ + 2pq cos j 



> III. 



. a . fi\ 



To I, II, III will correspond three different kinds of geometry 

 which are, respectively, the non-Euclidean geometry of Lebat- 

 schensky and Bolyai, the ordinary geometry and spherical ge- 

 ometry. 



The different kinds of multiplications of points are investigated 

 assuming the distributive principle, and it is shewn that the most 

 general conditions are 



P 2 = Q z = j3 f a constant 

 and P.Q + Q.P=2/3 cosh in I, 



P.Q+Q.P = 2{3 in II, 



P.Q+Q.P = 2/3 cos0 in III. 



Different kinds of multiplication can be obtained by making 

 further special assumptions. These are, 



1st. Grassmann's outer multiplication which is (for two points) 

 identical with the vector multiplication of Quaternions. 



2nd. Grassmann's inner multiplication which is (for two 

 points) identical with the scalar multiplication of Quaternions. 



3rd. The associative Quaternion multiplication with the 

 following laws : 



QP~ l = cosh + i sinh i 2 = 1 in I, 

 QP' 1 = l + c0 i 2 = in II, 



Qp- 1 = cos0 + i sin t 2 = - 1 in III, 



t is in each case a specific constant peculiar to the line joining 

 Pand Q. 



The addition and multiplication of these quantities i is next 

 considered. It is shewn that if i t i 2 be the specific quantities 

 peculiar to lines making an angle and if re -pi t + qi 2 



then r 2 =p 2 + q 2 + 2pq cos 0. 



Also . ij' 1 = cos + sin 



where O 2 = — 1, 



and may be identified with the point at which the lines meet. 



