244 Proceedings of the Poyal Society of Edinburgh. [Sess. 
Then if this differential form is reduced to the form ad/3 (there is a big 
theory dealing with the reduction of Pfaff’s expressions) we have 
and therefore 
df3 
-“M, 
^ G dt’ 
ld(a^ _ d(a,f3) 
" cd{x,t)’ " d(y,z)> 
which are the same as your expressions, except that the electric and 
magnetic r61es are interchanged. 
‘‘ I have been so exclusively occupied with pure mathematics for the last 
four years that I can’t say whether any of your work has been anticipated 
or not. — Yours very truly, E. T. Whittaker.” 
Before this reply from Professor Whittaker reached him, William Brown 
was with the Expeditionary Force in France. In later letters to his parents 
he discussed the significance of the Relativity Theory, and on September 16 
asked them to send him, if possible, a copy of Minkowski’s Raum und Zeit. 
In acknowledging books received on October 15, he asked for news of the 
Relativitdts Prinzip. Before the lapse of another month this young life 
had passed from among us, and the world was the poorer by the loss of an 
intellect brilliant in mathematical power and promise. 
W. G. Brown’s notes on the investigation described above have not been 
recovered, but there is no doubt that his mind was taking a firm grip of 
the mathematical methods associated with the modern theory of Relativity. 
To show the extent to which he had mastered the quaternion method, I 
conclude with giving a brief account of his note on Mass as a Linear V ector 
Operator. It is an interesting and novel generalisation of Tait’s early work 
on the Rotation of a Rigid Solid (see Tait’s Quaternions, §§ 406 et seq .) : — 
Let u be the velocity of the body (more strictly of a definite point in the 
body), and let the operator /x take the place of the mass factor in ordinary 
dynamics, being a self-conjugate linear vector function whose axes are 
fixed in the body. Then the momentum is 
7] = jX(T, 
The force /3 is assumed to be given by the Newtonian law 
force = — (momentum), 
(It 
or 
Since the quantities /3 and fx are referred to axes fixed in space, it is 
necessary to consider the variation jj,. 
