87 
function of ayz, or of p=ix+jy+ kz, and subject as such to 
the operations dq, S.ad, S.(84; where 
S.ad=-(a se rer) 
ifa=ia +jb+hc, and the symbol S.( < is similarly inter- 
preted. 
These were among the chief elements of calculation em- 
ployed, in proving by quaternions the theorem above men- 
tioned of Dupin; of which one expression, in the quaternion 
calculus, is the following :— 
‘« Tf the three differential equations, 
S. vdp = 0, S. v'dp = 0, S. vv dp = 0, 
be integrable, and if S.vv'=0, then the supposition V. v‘dp = 0 
conducts to the equation S$. vvdy=0.” 
Another expression of the same theorem is as follows : 
“Tf S.vdv=0, S.vdv'=0, S.v" dv’ =0, 
and V. vv'v’=0, 
then APE GSP a (a a 
Tn this last formula, the symbol S. v'<q.v denotes a vector 
having the direction of -dy, if dp have the direction of v’; 
and the equation expresses, that if we thus move a little along 
the first surface in the direction of the normal to the second 
surface, the new or near normal to that first surface will be 
contained in the tangent plane to the third surface, and there- 
fore will intersect the old normal to the first surface: which 
is a form of the theorem of Dupin. 
Although not very closely connected with that well-known 
theorem, Sir W. R. H. wishes to add that another old form 
of his, for any three vectors, namely, 
V(V.yB.a)=yS8.Ba-BS. ya, 
has suggested to him this new symbolical result, 
DEE. =. V(V.y4.vj)=yS.4v- 48.0; 
