86 
Sir W. R. Hamilton, having been lately induced to con- 
sider, in connexion with the Calculus of Quaternions, the 
celebrated theorem of Dupin, respecting the character of the 
intersection lines of three systems of orthogonal surfaces, as 
lines of curvature thereon, stated that he had thus been led 
to perceive some symbolical results which he supposed to be 
new, and which seemed to him to be of sufficient interest to 
be submitted to the Academy. 
As long ago as 1846, he had proposed the notation,* 
id tere d 
Saiaancd 7 +k ae 
and had pointed out a theorem,} differing only slightly in its 
expression from the following : 
V.aV. By=yS.aB—-BS. ay; 
which may also be thus written, 
V.a(V. By)=8.aB.y-BS.ay, 
V.a(V.By)=S.Ba.y-BS. ay. 
The recent results just referred to have a remarkable sym- 
bolical resemblance to those comparatively old ones, since they 
admit of being written thus : 
or thus, 
Teese V.a(V.dv)=S.ad.v-d8.a0; 
PPD: 2: V.A(V.Bv) =8.B4.v-BS. dv; 
where <is not an ordinary vector, but a certain symbol of ope- 
ration, analogous to a vector, in its combinations with other 
symbols, and defined by a foregoing formula: while a and B 
are constant vectors, and v isa variable vector, regarded as a 
* See Proceedings of the Academy for July, 1846. 
+ See Philosophical Magazine for August, 1846. 
