234 Proceedings of Royal Society of Rdinhurgli. [sess. 
thing is given in Hamilton’s Elements (p. 438, Equation XXYII.), 
and in Tait’s Quaternions (p. 89, 2nd edition; p. 123, 3rd edition). 
Of course, this particular way of inverting ^ depends on the parti- 
cular semi-cartesian form into which it is thrown. The rare beauty 
of Hamilton’s method, however, lies in its generality. For this Mr 
Heaviside has apparently only sc offings. 
In § 171 of the same series of articles, Heaviside criticises some of 
Professor Tait’s methods of developing the quaternion calculus. 
His criticisms are equally valid against Hamilton’s own methods. 
Regarding these he says : — 
“ The reader is led to think that the object of the investigation is 
to invert a linear operator — that is given [p = cjScr] to find [o- = 
But if this were all it would be a remarkable example of how not 
to do it. For the inversion of a linear operator can be easily effected 
by other far simpler and more natural means. The mere inversion 
is nothing. It is the cubic equation itself that is the real goal. 
The process of reaching it is simplified by the omission of inverse 
operations.” 
But what says Tait in § 174 of the 3rd edition (§ 162 of the 2nd) : — 
“ It is evident from these examples that for special cases we can 
usually find modes of solution of the linear and vector equation 
which are simpler in application than the general process of § 160. 
The real value of that process, however, consists partly in its 
enabling us to express inverse functions of <jf>, such as (cj^ - g)~^ for 
instance, in terms of direct operations, a property which will be of 
use to us later ; partly in its leading us to the fundamental cubic 
whose interpretation is of the utmost importance with 
reference to the axes of surfaces of the second order, principal axes 
of inertia, the analysis of strains in a distorted solid, and various 
similar enquiries.” 
Could words be plainer or more emphatic? The most lenient 
hypothesis is, that our self-appointed critic has not really read 
Tait’s Quaternions. 
Then, as to the much vaunted “ simplified ” process of reaching 
the cubic, what does it amount to ? It consists in the highly 
original trick of writing 
m~V jjiv = <i>'V 
m(j> fxv = \ cj^gcjiv ! 
instead of 
