114 Proceedings of Royal Society of Edinburgh. 
This is the equation of equilibrium of an elastic body (solid or fluid) 
subjected to an external force and couple per unit volume of the 
unstrained body 3 and 931 respectively. For the corresponding 
equation of motion we have to equate the left member of this 
equation to Dp', where D is the density of the body at the point 
in the standard state. 
In the case considered by Thomson and Tait (Nat. Phil., 
App. C.), both § and 93f are zero, and our equation takes the very 
simple form 
p/SV^awA =0 (43) 
the exact quaternion equivalent of their three equations (7).* 
In connection with equations (41) and (42), it should be observed 
that 
to) — - JVSKx • • • . (44) 
where t has the same meaning as in equation (38). 
The next few applications will he of an extremely simple nature, 
and will he confined to work already well known in its Cartesian 
form. 
We now assume, as is usual, that the strains are small, and that 
there is no external couple, and therefore no stress couple. We 
deduce our equations as particular cases of the above, though, of 
course, if the present were our sole object, we could adopt a much 
simpler process. We may put q— 1, <£ = <£ = zs, 2if/ = x + X> an( ^ 
o) — o). Also = 0, and .-. 6 of equations (36) 
= 0. Thus equation (36) gives 
» = (45). 
It is worth while giving one of these for comparison. I may remark that 
many of the results arrived at above, although quite simple enough in their 
quaternion form to be manageable, and therefore useful, become so extra- 
ordinarily complicated when translated into Cartesian notation as to be utterly 
unmanageable and useless. The equation referred to is 
d 
{■£(=*>)■ 
dw da 
dw da \ 
dx 
*~db Hz + 
dc dy J 
d 
dy 
f dw da dw 
l dy + da 
da dw i 
dz dc ' 
ijD 1 )} 
-\ 
dz 1 
f dw da dw 
l dQ~dz + da 
da dw 
dy + db 1 
(£+*)} 
