176 ON SOME QUARTERNION TRANSFORMATIONS. 



These forms simplify still further when in fluid-motion <r 

 is the velocity at any pointy and Y^(r=2p gives the rota- 

 tion at that point, in which case S\fp=o. 

 We have then 



Vv(V/jc7) = (Spv)o-— (So-vV — s v^ • P> 



and 



V(Spo-) = Vvpo-+ (Spv)o"+ (So-v)p- 



Again, Helmholtz's notation for the form of a gives 

 a=^[(fi + Qi), where ^ is a scalar and Sv« = o; and these 

 formulae 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 t, we should 

 have 



n. 



These results are very useful in obtaining modified 

 forms of Greenes theorem. 



The general form of the theorem is 



JSvi|r6fe=:j'ST|rv . ds, 



where -x^ 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. 



Case I. Let i/r=V(po-). 

 Then by (V) 



and we get 



^^^pardv—^Sp'^(Tdv=^SYp(T.vds=^Spavds. . (VII.) 



