124 
An immediate result is, that the ‘‘ Equation of the Wave” may be 
symbolically expressed as follows :— 
0= Sp" ($-p*y'p"; (b) 
or, by an easy transformation,— 
1=Sp (p°— $7)" p. (b’) 
Of these formule, likewise, the agreement with known results (in- 
eluding one of his own) has been verified by Sir W. R. H., who has also 
found that it is as easy to return, in the quaternion calculations, from 
the wave to the index-surface, as it had been to pass from the latter to 
the former: the only difference worth mentioning between the two pro- 
cesses being this, that when we interchange u and p, in any one of these 
formule, we are at the same time to change the symbol of operation, >, 
to the inverse operational symbol, G*. 
3. From the expression (b), by the introduction of two auxiliary and 
constant vectors, «, x, such that (as in the Lecture above cited) the fol- 
lowing identity holds good :— 
Spdp = ( 
Sir W. R. H. has lately succeeded in deducing, in a new way, a less 
symbolical, but more developed, guaternion form for the Equation of 
the Wave, which he communicated in 1849 to a few scientific friends, 
and which he wishes to be allowed to put on record here: namely, the 
equation, 
poms 
Bene) 
(2-2)2= {8 (e—«) p}? +(L¥ip+ TV ep); (c) 
which exhibits the physical property of the two vectors, +, x, as lines of 
single ray-velocity ; and is also adapted to express, and even to suggest, 
certain conical cusps and circular ridges on the Biaxal Wave, discussed 
many years ago. 
In the course of a recent correspondence, on the subject of the qua- 
ternions, with Peter G. Tait, Esq., Professor of Mathematics in the 
Queen’s College, Belfast, Sir W. R. Hamilton has learned that Professor 
Tait has independently arrived at this last form (c) of the Equation of 
Fresnel’s Wave; and he hopes that the method employed by Mr. Tait 
will soon be, through some channel, made public. In the meantime he 
desires to add, for himself, that he is not to be understood as here offer- 
ing any opinion of his own on the rival merits of any physical hypotheses 
which have been proposed respecting the directions of the vibrations in 
a crystal, or other things therewith connected; but merely as applying 
the CatcuLus of QuATERNIONS, considered as a MatHEmaticaL ORGAN, to 
the statement and combination of a few of those hypotheses, especially 
as bearing on the WAVE. 
(To be continued.) 
