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VI. — On the Degree of the Surface reciprocal to a given one. By the Rev. 

 George Salmon, Felloiu of Trinity College, Dublin. 



Read November 30, 1855. 



I. — General Theory. 



It is my intention in the present memoir to lay before the Academy the ex- 

 tension and completion of a theory of reciprocal surfaces, the first outlines of 

 which I published some years ago in the " Cambridge and Dublin Mathema- 

 tical Journal" (vol. ii. p. 65, and vol. iv. p. 187). I there showed how to cal- 

 culate the degree of the reciprocal of a surface having an ordinary double line ; 

 it remains now to show what the degree will be when the surface has likewise a 

 cuspidal line (that is to say, a double line, the two tangent planes at every point 

 of which coincide). I purpose next to examine the nature and number of those 

 singular tangent planes to a surface which give rise to multiple points and lines 

 on the reciprocal surfoce, and thus to show how it is that the degree of the 

 reciprocal of that reciprocal coincides with the degree of the original surface. 

 We shall thus obtain results analogous to the well-known theorems of M. 

 Plccker for the case of curves. Lastly, I purpose to apply this theory to the 

 case of developable surfaces, and to show how it is that the degree of the reci- 

 procal of a developable reduces to nothing. I may mention that the substance 

 of the present paper was prepared for publication in the year 1 849, though various 

 causes have prevented its being published until now. 



I use the following notation for the following quantities, which will come 

 under consideration in the discussion of the problems which it is proposed to 

 investigate, and which may be regarded as the ordinary singularities of sur- 

 faces : — 



VOL. xxra. 3 p 



