400 Mr. Oliver Heaviside on the 
ternions by simply defining a scalar product to be so and so, 
and a vector product so and so. The investigation is thus a 
Cartesian one modified by certain simple abbreviated modes 
of expression. 
I have long been of opinion that the sooner the much needed 
introduction of quaternion methods into practical mathematical 
investigations in Physics takes place the better. In fact every 
analyst to a certain extent adopts them: first, by writing only 
one of the three Cartesian scalar equations corresponding to 
the single vector equation, leaving the others to be inferred ; 
and next, by writing the first only of the three products which 
occur in the scalar product of two vectors. This, systematized, 
is I think the proper and natural way in which quaternion 
methods should be gradually brought in. If to this we further 
add the use of the vector product of two vectors, immensely 
increased power is given, and we have just what is wanted in 
the three dimensional analytical investigations of electro- 
magnetism, with its numerous vector magnitudes. 
It is a matter of great practical importance that the notation 
should be such as to harmonize with Cartesian formule, so 
that we can pass from one to the other readily, as is often 
required in mixed investigations, without changing notation. 
This condition does not appear to me to be attained by Pro- 
fessor Tait’s notation, with its numerous letter prefixes, and 
especially by the —S before every scalar product, the nega- 
tive sign being the cause of the greatest inconvenience in 
transitions. I further think that Quaternions, as applied to 
Physics, should be established more by definition than at 
present ; that scalar and vector products should be defined 
to mean such or such operations, thus avoiding some extremely 
obscure and quasi-metaphysical reasoning, which is quite 
unnecessary. 
The first three sections of the following preliminary con- 
tain all we want as regards definitions ; most of the rest of the 
preliminary consists of developments and reference-formule, 
which, were they given later, in the electromagnetic problem, 
would inconyeniently interrupt the argument, and much 
lengthen the work. 
Scalars and Vectors.—In a scalar equation every term is a 
scalar, or algebraic quantity, a mere magnitude ; and + and 
— have the ordinary signification. But in a vector equation 
every term stands for a vector, or directed magnitude, and + 
and — are to be understood as compounding like velocities, 
forces, &c. Putting all vectors upon one side, we have the 
general form 
A+B+4+C+D4...=0; 
