2 Proceedings of the Roijal Irish Academy. 



On this supposition, 



p 



1 



p' 



1' 



p 



9. 



p 



1 



= P'i - If't ^^t ^0* pi ~ P'l 5 

 = ^g- - 2^ = 2 F. F^ Vq, and 



= 'P1-pq_-=^' 



It is also obvious that if a; is any scalar, 



P ? 

 / 9! 



p xp + q 

 p' xp' + q' 



but not = 



p p 

 q q 



P i 



xp + p' xq-^ q' 



Thus the columns may be treated as in ordinary determinants with 

 scalar constituents ; but it is not lawful to treat the rows in this 

 manner. The former of these processes is consistent with the con- 

 vention that the order of the constituents shall follow the order of the 

 rows ; the latter violates this convention. 



3. Multiplication of a quaternion and a scalar determinant. — Again, 



P 9 

 p'q' 



X y 



px + qy px' + qy' 

 p'x + ^y p'x' + ^y' 



px + qx' py + qy' 

 p'x + q'x' p'y + q'y' 



the p and q being here, as elsewhere in this Paper, quaternions, and 

 the X and y being scalars. Similar processes hold for determinants of 

 any order. 



Further, it is easy to see that, \i p = w ^-ix -^jy + Jcz, with similar 

 expressions for the dotted letters, 



p p' p" 





1 ij k 





w X y z 



p p' p" 



= 



I i j k 





w' x' y' %' 



p p' p" 





1 i j k 





w" x" y" %" 



4. Determinants with identical rows. — As geometrical examples, 

 observe that if a, y3, y, 8, &c., are vectors, 



a p 



a /3 

 a (3 y 

 a (3 y 

 a ^ y 



= 2ra)8: 



2 (aFJSy + )8 Fya + yVa{3) = 6Sa{3y; 



