668 Proceedings of the Boyal Society 
and 5 is therefore to be found by the solution of a biquadratic 
equation, as in Proc. R. S. E., 1870, p. 316. It is evident, indeed, 
from the identical equation 
S . <r-p'pp = S . ppf pa- 
th at the operator p'p is self- conjugate. 
In the same way 
9? (. ) = 2 — 1 S 2 (g ( )q~ 1 )q 
or 
2 (w'p) 2 _1 = (2P2 -1 ) = <p'<p (qpq- 1 ) 
which show the relations between pp', p'p, and q . 
To determine q we have 
<pp.q = q*rp 
whatever be p, so that 
or 
which gives 
S.Yj(p -*)p= 0, 
S . p (p' — sr) Yg = 0 , 
(p' - w) = 0 . 
The former equation gives evidently 
V# || Y. (9 - a (p - 5) /? 
whatever be a and /? ; and the rest of the solution follows at once. 
A similar process gives us the solution when the rotation precedes 
the pure strain. 
4. On the Motion of Bigid Solids in a Liquid circulating 
Irrotationally through Perforations in them or in any 
Fixed Solid.* By Sir William Thomson. 
1. Let if/, p, ...be the values at time t, of generalised co-ordi- 
nates fully specifying the positions of any number of solids mov- 
able through space occupied by a perfect liquid destitute of rota- 
tional motion, and not acted on by any force which could produce 
* The title and first part ($£ 1 ... 13) are new, The remainder (§§ 14, 15) 
was communicated to the Royal Society at the end of last December. — W. T. 
September 26, 1872. 
