1890 - 91 .] Mr A. M'Aulay on Quaternion Differentiation. 103 
f<f>(d P )=/f<f>(VUv& )ds (2), 
JJW vds ==///<]> Ad* ...... (3). 
The last equation shows at once, as pointed out in the paper already- 
referred to, that the stress <j> (when <f> has the particular form of a 
vector function of a vector) causes a force per unit volume <f> A . 
Before considering the properties and applications of V in this 
extended form, I will now introduce all the innovations that I 
propose. 
A large class of differentiations can he included under a symbol 
somewhat analogous to V . Suppose we have a linear vector func- 
tion, «f>, of a vector depending on the nine scalars, a, b , c, a\ b\ c, 
a", b'\ c", by means of the equations — 
<fii = ai + bj + 6k 
<f)j =a'i +b'j +ck 
tf>k = a"i + b'j + c"k 
Then a, 6, c, &c., can be called the coordinates of <f>. Again, 
P, Q, B, L, M, N may be called the coordinates of the self-conjugate 
linear vector function, zf, of a vector, if 
tffi = P i + N; + Mk 
zsj = Ne + Q j + L k 
rsk = M i 4- Lj + B& . 
Now, just as, if u, v, w are the coordinates of an independent vector 
o-, o-V may be defined as a symbolic vector, whose coordinates are 
~t -- > ; so if a, A c. a\ b\ c\ a\ \ c' 9 be the coordinates of 
du dv aw 
an independent linear vector function, <£, of a vector, may be * 
defined as a symbolic linear vector function, whose coordinates 
are — 
d d d d d d d d d 
da 5 db 5 dc 9 da' 9 db ' 5 dc' 9 da ' 5 db" 5 dc" ’ 
If 7tf be an independent self-conjugate linear vector function of a 
* I use the inverted D to suggest the analogy to Hamilton’s inverted A. It 
is advisable to write o-V, &c., instead of v<r, &c., as the numerical 
suffixes must be much more frequently introduced than the symbols <f> or <r, 
which may be very frequently understood, and it is not advisable that both 
the literal and numerical suffixes should be on the same side of the d or v. 
