532 
Proceedings of the Royal Society. 
easily seen to be equivalent to the kiuematical problem of finding 
a displacement which shall produce no compression, but shall 
produce a rotation whose vector axis itself corresponds to a dis- 
placement without compression. 
The nature of the difficulty is also easily seen in another way ; 
for, when we try to find the conditions of integrability of such an 
equation as 
V . Xdp. = dv , 
we may, of course, make the assumption 
d/x = 
where the coefficients of <f> are functions of p. This gives at once 
S adp. — S. <j)'adp , 
so that 
V.V<£'a = 0 
whatever constant vector be a. 
Suppose this satisfied, we have the farther condition 
dp = dv , 
or 
S.<f>'Y (aX) dp = S adv , 
so that, whatever be a , 
V.V<£' (YaX) = 0 
Taken in conjunction with the former condition, this shows that 
V may here be considered as operating on X only. 
In this very particular case, however, we find at once that X must 
be constant, and that 
dp. = <f>dp = idu -f jdv + kdw . 
