291 
terly Memoir on Surfaces of the Second Order, which has 
been published by Professor Mac Cullagh in the Proceedings 
of this Academy. (See Part VIII., page 484.) 
The length to which the present abstract has already ex- 
tended, prevents Sir William Hamilton from offering on the 
present occasion any details respecting the processes (analo- 
gous in some respects to the calculi of variations and partial 
differentials) by which he applies the principles of his own 
method to investigations respecting surfaces and curves in 
space, or to physical problems connected therewith ; he.de- 
sires, however, to mention here that, in investigations respect- 
ing normals to surfaces, he finds it convenient to employ a 
new characteristic of operation of the form 
(s.dp).d=a, (48) 
in order to obtain from a scalar function of a variable vector 
p, a new variable vector v which shall be normal to the locus 
for which that scalar function is constant ; and that the fol- 
lowing more general characteristic of operation, 
pinta @ d 
alae ds (49) 
in which z, y, z are ordinary rectangular coordinates, while 
i,j, k are his own coordinate imaginary units, appears to him 
to be one of great importance in many researches. ‘This will 
be felt (he thinks) as soon as it is perceived that with this 
meaning of <q the equation 
has) Mapes i, OD 
is satisfied in virtue of the fundamental relations between his 
symbols #, 7,4 ; which relations give also, as another result of 
operating with the same characteristic, this other important 
symbolic expression, which presents itself under the form of 
a quaternion : 
