880 Proceedings of Royal Society of Edinhurgh. [june 4 , 
Do-Vcr — VDo-cr = Do-V<t — (VjDo-j^o- + Dg-Vcr) 
= — Vj^Do-jOr = VjScTjV . cr 
— (Vo-)^ - S . VjO-^v • o- - Vcr . SVa + S VVj . ctjOt . 
The hydrokinetic equation is 
D,<r = v(P-f), 
SO that V . VDo-o- = 0 ; 
or, by the above transformation, 
V. D^Vcr = V(ViSo-iV.o-) 
which is the equation treated by Cayley. 
It is worthy of note that the right-hand member may be written 
as 
V(Vo-)2 - S . ViO-^V . O- - Wo- . SVo- 
because S VV^ . Vo-j^o- = 0 identically. 
If we now introduce the equation of continuity 
SVo- = 0, 
we have (as in the abstract referred to) 
Dq-Vo- = - S . Vjo-j V . a- = 8v<rO- , 
with the further result 
- V^(p = (Vcr)2 + SVVj . So-o-i . 
(h) The second note contains additions, of which 
fffY . VVo-Tc/s = #(tSo-Uv - o-SrUv)rf5 
may be given as a specimen, to the paper on Quaternion Integrals 
printed in abstract in Proc. Roy. Soc. Edin., vii. 318, 784. 
PRIVATE BUSINESS. 
Mr C. M. Aikman and Mr George Williamson were balloted for, 
and declared duly elected Fellows of the Society. 
