784 Proceedings of the Royal Society 
and alter the value of /x, we have by taking a small, 
+ n 2 + fx 
which, with n = 0, is the usual equation for sound, provided the 
particles repel one another. Of course we can easily extend the 
investigation so as to include the more complex cases where the 
mutual actions of all the poles are taken into account. The result 
is not altered in form ; hut it might he curious to inquire whether 
the retention of n 2 in the equation might not give some hints as to 
the formation of a dynamical hypothesis of the action of transparent 
solids on the luminiferous ether. This, however, I cannot enter 
upon at present. 
4. On Some Quaternion Integrals. Part II. By Professor 
Tait. 
( Abstract .) 
Commencing afresh with the fundamental integral 
fjf S . Xcrds = Jf S . B vcrds , 
put 
cr = u/3 
and we have 
J]f(f> . [dV)uds —Jfu S . /3Yv ds ; 
from which at once 
JffVuds =JfvXJvds, . . . (a), 
or 
fff^rds^ffVv.Tds. . . . (5). 
Putting u x t for t, and taking the scalar, we have 
JJJ (S(tV) . u x + u x S. Vt) ds — JJ tqS . TJvr ds 
whence 
fff (S (tV) «r~+ crS . Vt) ds = ff S . Uvt ds . . (c). 
As one example of the important results derived from these 
simple formulae, I take in this abstract the following, viz.:— 
JfY . (Y . cr- TJv) rds ^jfrS.Vvrds - JfUvS . nds } 
