1892 - 93 .] Prof. Knott on Innovations in Vector Theory. 221 
products, it follows that consistency demands that aa or a? should 
be put equal to negative unity. There seems to me to be absolutely 
no way out of this conclusion in any vector analysis in which the 
product of two perpendicular unit vectors is taken to be a third 
unit vector perpendicular to their plane, unless it be assumed that 
a vector operating a second time reverses its action, and undoes the 
effect of the first operation — in short, is a recigprocating versor. 
Nevertheless Macfarlane, in his Princigoles of the Algebra of 
Physics, insists that Hamilton had no rational ground for putting 
the square of a unit vector equal to negative unity. This was 
exactly O’Brien’s difficulty forty years ago. Heaviside takes the 
same view. Professor Gibbs appears to side with them by arguing 
that - Sa/1 and not is the quantity we should fix our attention 
on. He, however, does not admit into his calculus the complete pro- 
duct at all, even in the cases in which one of the parts vanishes. 
For example, with i j h, in their usual significance, he defines 
i.i ( = - Sw) =1, i xj( = Yij) = Zq &c. &c., 
but the product ij has no explicitly recognised place in his system. 
On the other hand, both Heaviside and Macfarlane boldly write 
ij = h, i^= -fl, &c., ifec. 
Now let us take what is common to quaternions and to these 
other vector systems, namely, the well-known set of equations con- 
necting mutually rectangular unit vectors : — 
ij=.h= -ji 
jk = Z = - kj 
ki ^ j = — ik 
and let us form the product of the three vectors i, i-\-j , j. 
By one mode of association 
+i)i = + ij)j = iy + kj = -f iy - l 
By another mode of association 
iy +i)i = +/) H- ip = - j -f ip. 
Now it is evident at a glance that these cannot be the same unless 
P=j'^= -1. In other words, for the associative law to hold, we 
must of necessity take the squares of ij k to be negative unity. If 
* See also Kelland and Tail’s Introduction to Quaternions (chapter hi.) for 
an even simpler proof. 
