Surface reciprocal to a ijiven one. 487 



two points on each of the double lines which affect the number of cuspidal 

 edges. The number of stationary tangent planes continues = as before ; but 

 in any cone having no stationary tangent planes, if we diminish the degree of 

 the cone by any number, and the number of its cuspidal edges by three times 

 the same number, the degree of its reciprocal is not altered. 



I am not able in general to apply this theory to the next simplest class of 

 ruled surfaces, viz., those generated by a right line resting on one right line and 

 two curvilinear generatrices of degrees fx, fx respectively. The degree of the 

 sirrface will be 2/x/i', and the right line will be multiple of the order fi/x, and 

 the curves of the orders /x', ju, respectively. There will be a double curve of 



the degree [at least ?] -, which each generating line meets 



in {fx — \) (/i' — 1 ) points. A certain number of generators are also double lines 

 (see "Cambridge and Dublin Mathematical Journal," vol. viii. p 46). I can 

 satisfactorily explain the case where fj. = fi' =2, but, as I have said, I cannot 

 completely account for the general case. 



I have also examined the ruled surface generated by a right line resting 

 twice on a given curve of degree ju, and once on a right line. This will have 

 the curve and right line for multiple lines, and, in addition, a double curve of 



the degree [at least ?] ^-^ 9 a — ^ ^^^ satisfactorily account 



for the case fx = S, and also when the curve is the intersection of two surfaces 

 of the second degree ; but I do not know the theory of the general case. 



Note. — February 12, 18.^7. — I have just received the " Quarterly Journal 

 of Mathematics," No. 5, which contains a paper by Professor Schlafli, going 

 over some of the ground traversed in the present memoir. In particular, 

 Dr. Schlafli obtains the values given (p. 477) for /, n', b', c', ^, and 7'; — of 

 these I have already published n', c' (" Cambridge and Dublin Mathematical 

 Journal," vol. iv. p. 188), and given methods which lead to the determi- 

 nation of J', ^, 7' ("Mathematical Journal," vol iv. pp. 119, 260). 



