1872.] Mr. W. Spottiswoode on the Contact of Surfaces. 



179 



February 22, 1872. 



WILLIAM SPOTTISWOODE, M.A., Treasurer and Vice-President, 



in the Chair. 



The following communications were read : — 



I. " On the Contact of Surfaces." By William Spottiswoode, 

 M.A., Treas. R.S. Received January 18, 1872. 



(Abstract.) 



In a paper published in the Philosophical Transactions (1870, p. 289), 

 I have considered the contact, at a point P, of two curves which are co- 

 planar sections of two surfaces (U, V) ; and have examined somewhat in 

 detail the case where one of the curves, viz. the section of V, is a conic. In 

 the method there employed, the condition that the point P should be 

 sextatic, involved the azimuth of the plane of section measured about an 

 axis passing through P ; and consequently, regarded as an equation in the 

 azimuth, it showed that the point would be sextactic for certain definite 

 sections. It does not, however, follow, if conies having six-pointic contact 

 with the surface U be drawn in the planes so determined, that a single 

 quadric surface can be made to pass through them all. The investigation 

 therefore of the memoir above quoted was not directly concerned with the 

 contact of surfaces, although it may be considered as dealing with a 

 problem intermediate to the contact of plane curves and that of surfaces. 



In the present investigation I have considered a point P common to the 

 two surfaces U and V, an axis drawn arbitrarily through P, and a plane 

 of section passing through the axis and capable of revolution about it. 

 Proceeding as in the former memoir, and forming the equations for contact 

 of various degrees, and finally by rendering them independent of the 

 azimuth, we obtain the conditions for contact for all positions of the 

 cutting plane about the axis. Such contact is called circumaxal ; and in 

 particular it is called uniaxal, biaxal, &c, according as it subsists for one, 

 two, &c. axes. If it holds for all axes through the point, it is called 

 superficial contact ; and in the memoir some theorems are established re- 

 lating to the number of sections along which contact of a given degree 

 must subsist in order to ensure uniaxal contact, as well as to the con- 

 nexion between uniaxal and multiaxal contact. At the conclusion of § 3 

 it is shown that the method of plane sections may, in the cases possessing 

 most interest and importauce, be replaced by the more general method of 

 curved sections. 



In the concluding section a few general considerations are given re- 

 lating to the determination of surfaces having superficial contact of various 

 degrees with given surfaces ; and at the same time I have indicated how 

 very much the general theory is affected by the particular circumstances of 

 each case. The question of a quadric having four-pointic superficial con- 



