577 
1907-8.] Algebra after Hamilton, or Multenions, §11. 
Thus the linear quaternion function of a quaternion, of which Professor 
J oly has made such admirable use, is a multenion expressible by four fictits. 
These fictits may be taken as the linear quaternion functions (quaternion 
linities) 
»’( )J( ), K )h K )h 
or if law A is to hold, as 
•( )> i( \ M )ij(- 1), k{ );V( - 1). 
These are obtained by putting below : — 
c = 1, n = 2 ; q = ij( - 1), e 2 =j ; and therefore 67 0 = kj{ - 1), 
and they may be taken to illustrate the meanings of the proof of the 
general proposition, now to be given. 
Law A could be assumed and the imaginary made use of, but it is much 
simpler to violate law A and to connect therewith when required by aid of 
J( — 1). It should be remarked that taking i ls /( — 1) instead of q as a 
fictit changes the meaning of K to KI P but does not change the meaning of 
any other symbol of replacement such as the following and combinations 
of them, 
P,Q, I P I 2 . . . . 
According as n is even or odd, let its value be 2c or 2c + 1, and let the 
fictits of the given multiplex be denoted by e v e 2 , .... e 2c , e, the last being 
omitted when n = 2c. Let 
1 = er = - e 2 2 = € 3 " = . • • . — e 2c 2 = e 2 . . . (37) 
Let ^o =e i e 2 . • . . e 2c . . . . . . (38) 
so that when n = 2c, is the product of the fictits, and when n = 2c + 1 , zsr 0 6 
is the product. 
It is convenient to note that (37) and (38) give 
1 = = (<1< 2 ) 2 = (*iW =....= ^T 0 2 = (*V^ 2 . • ( 39 > 
Define the multilinities A l5 X 2 , . . . . X 4c+2 (omitting the two X 4c+1 , A 4c+2 
when n — 2c) by the equations 
Ai e 4 ( ), A 2 = e 2 ( ),...., A 2c = e 2c ( ) \ 
A 2c+ i = tTT 0 ( )e 2 , A 2c+2 = £T 0 ( )e 2 , . . . ., A 4c = CT 0 ( )e 2c V . . (19) 
A 4c+ i= P( ), A 4c+2 = CT u eP( ) ’ 
It is a quite simple matter to verify that these linities are all anti- 
commutative with one another ; to take account of A 4c+2 it is perhaps easiest 
first to verify that A 4c+2 is anti-commutative with A 4 c+ i, and then to verify 
vol. xxviii. 37 
