89 
whence we may conclude that a principal axis remains a 
principal axis. 
We may, therefore, now suppose that we have chosen 
surfaces $ intersecting in principal axes of strain at the 
points under consideration, as the surfaces to which we are 
referring the circumstances of motion, and we must show 
that these sets of surfaces will indicate the principal axes at 
other points. We might, perhaps, conclude this from the 
fact that any scalar operation of Y08 1 produces a zero 
result ; but we will show the surfaces of reference intersect 
at right angles at consecutive points. 
In the case considered a = fav0i> &c. Sa/3 = Sv^iV^a = 0, <fcc. 
^= s ^ +s “| 
“ + * lS 4a V ^ + 
= 0 
with similar results for the other expressions. There is, 
therefore, one and only one set of surfaces which move 
with the fluid and cut everywhere orthogonally, and these 
intersect in the principal axes of strain at every point. 
It will be noticed that the quantity fa written above may 
H / , p a \2 
be shown = ^ 2 = , and that H 2 = fa 2 (T «) 2 = fa 2 fa 2 fa 2 . 
When using these surfaces as those of reference ex- 
pressions, such as Vp undergo considerable simplification. 
If in (9) £ had been a principal axis, we should have 
obtained 
- = JSp^'(Vp8 + S 1 ) = j(Vp£) 2 = J(Tp)(T£) 2 sin 2 0, 
and as will be seen later p is one of these principal axes 
only when VpD*p = 0, or when p moves parallel to itself. 
2. In certain cases we may simplify considerably the ex- 
pressions which we have found. 
