668 
Proceedings of the Royal 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 . a~(p'(pp = S . p (ft (per 
that the operator ft (p is self-conjugate. 
In the same way 
n' ( ) = 1 ( )q~ 1 )q 
or 
q (<pftp) q - 1 = ^ {qpq- 1 ) = ftp ( qpq -1 ) 
which show the relations between pft, ftp, and q. 
To determine q we have 
<pp . q = q w p 
whatever be p, so that 
or 
which gives 
S . Yq (yp — s) p = 0 , 
S . p (yft - sP) Yq — 0 ? 
(p' — sr) Yq = 0 . 
The former equation gives evidently 
Yq || Y. (<p - sr) a (<p - sr) ft 
whatever be a and ft) 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 Rigid Solids in a Liquid circulating 
Irrotationally through Perforations in them or in any 
Fixed Solid.* By Sir William Thomson. 
1. Let if/ 9 9 , ...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. 
