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from the former by the formula n =m (m-1). But this led to 
the paradox, that if we formed by the same rule the degree 
of the reciprocal of the reciprocal, instead of falling back on 
the number m, as we plainly ought, we should obtain a much 
larger number [(m?—m) (m?-m-1)]. The difficulty was ex- 
plained by showing that the degree of the reciprocal of a curve 
is diminished when the curve has multiple points; and the 
full examination of the subject showed that a curve of the m 
degree has in general a certain determinate number of points 
of inflexion and double tangents, each of which gives rise to a 
multiple point on the reciprocal curve. 
‘‘ The corresponding problems for surfaces were, I believe, 
first investigated in a paper which I contributed to the ‘Cam- 
bridge and Dublin Mathematical Journal’ in the year 1846, 
in which I gave the first outlines of a theory, the completion 
of which I now lay before the Academy. 
‘¢ In the following paper I first investigate the degree of the 
reciprocal of a surface of the m'* degree, and examine how that 
degree is affected when the surface has multiple points or lines. 
‘* The first application of the theory is made to the case of 
developable surfaces. The reciprocal of a developable is a 
curve of double curvature, which is to be considered as a sur- 
face of degree (0). It furnishes then a test of the theory to 
examine whether it explains why, when the surface is a deve- 
lopable, this reduction takes place in the degree of its reci- 
procal. And this explanation is successfully obtained. 
‘*T next show that a surface has a number of stationary and 
double tangent planes, whose points of contact lie on a certain 
locus, the degree of which is investigated. The surface has 
also a certain determinate number of triple tangent planes. 
Every one of these multiple tangent planes gives rise to a 
multiple point on the reciprocal surface. 
‘In the next place, having in the preceding section deter- 
mined the number of multiple points and lines on the reci- 
