119 
But r may be written aa-\-h[5+cy, where a, j3, y are rect- 
angular unit vectors. D in this case becomes identical with 
V, and the preceding form may be deduced. 
The more general form is occasionally useful. 
From I. we may by taking scalar and vector parts get 
Sv(po-) = Svpo" - Svpo- III. 
Vv(po-) = Yypa - Ypycr + 2(Spv)o- YV. 
whence we may deduce by putting 
Vvo-^ = 2 Yvo-(t + 2(So-v)o- 
We may also deduce from (1) S values for 
V(Spo-). To obtain forms more generally useful than the 
more general forms put Syp^O in 
yV(po-) = 5v(po- - ap) = S. vpo- - S.py/T + (Spy)o- - {So-y)p 
+ (Syp)or - p(Syx>) 
and we get 
Sy Vp<r = S(vpo’ - pyar) V. 
Vy Vpo- = (So-y)<r - (So-y)p VI. 
These forms simplify still further when in fluid motion a is 
the velocity at any point and Yy<7=2p gives the rotation at 
any point. 
Again, Helmholtz’s notation for the form of g gives 
cr=y(<^ + w) where is a scalar and Sya»=0, and these for- 
mulae are applicable in the reduction of the equations. 
A slight adaptation of this method enables us to prove 
that if p and q are quaternion functions of r, we should 
have 
y{pq) = ypq + Kpyq + 2(Spy)q 
II. 
These results are very useful in obtaining modified forms 
of Green’s Theorem. 
The general form of the theorem is 
f Sy\pdv = JSxpv.ds. 
where i// is a single valued vector function in simply con- 
nected space, dv an element of volume, ds of its bounding 
surface, and v the unit vector normal to ds drawn outwards. 
