1907-8.] Algebra after Hamilton, or Multenions, §§ 8, 9. 553 
Now, without exception, that is whatever be <j> and y, as long as [x] + 0, 
we can show from (22) § (7) that 
x^x^-xK = x^x^-xK = a x\ + (x x i)'> etc -, 
and therefore that the 7i-tics of <fi and X0X -1 h ave the same roots. 
9. Complementary Replacements, or the principle of Duality.— 
By the principle of duality I mainly mean that there are modes of 
translating properties of plane areas, their intersections, magnitudes, 
orientations, etc., into properties of straight lines, their intersections, 
magnitudes, orientations, etc. In quaternion geometry we have the 
striking applications of this principle, (1) that vector areas may be 
mathematically identified with vector lines, and (2) that a unit vector may 
be identified with a quadrantal versor. 
In our present subject the principle, no doubt, has very many applica- 
tions, and one is, so to speak, on the surface. The mere fact that a multit 
has a definite complementary multit associated with it suggests that many 
of the properties of multenions are identical with properties of the comple- 
mentary multenions. We are led naturally to inquire whether, with our 
technical use of “ replacement,” a multenion may be replaced by its 
complement (it may not), or whether there is any replacement intimately 
connected with complements (there is). 
Before entering on this matter of complementary replacements, consider 
an allied problem. 
Corresponding to the equation [(1) § 6] 
q.S n a {c) a {n ~ c) = 
or q. S n a< c )a^- c ) = ^ 2 S c a( c tS^.S n _ c a<”- c )) . . . . (1) 
there is clearly another equation in which occur f3 v /3 2 .... , where /3 1 . . . . 
are arbitrary (n — l) th order multenions, in place of ct_ p a_ 2 , .... (or let us 
say cq, a 2 , . . . .) which are first order multenions (or fictors). . Let us find 
this other equation. 
To avoid undue complexity I find it necessary to define as follows : 
iSj, means S k ivhen 1 is even, and means S n _ k when 1 is odd. In the form 
of an equation this is 
iS,=i[i+(-y]s,+j[i . . . . (2) 
Change the a of (1) into vy/3 so that, as is required, /3 is an arbitrary 
(n— l) th order multenion. Here w as usual stands for a product, in any 
sequence, of q, i 2 , . . . . i n , (n fictits constituting a fictorplex to which cq, 
a 2 , .... a n belong, whether or not cq, a 2 , .... a n are independent). 
