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XX. On Mr. Spottiswoode’s Contact Problems. By W. K. Clifford, M.A., Professor 
of Applied Mathematics and Mechanics in University College , London. Commu- 
nicated by W. Spottiswoode, M.A., Treas. & V.P.R.S. 
Received June 19, — Read June 19, 1873. 
The present communication consists of two parts. 
The first part treats of the contact of conics with a given surface at a given point ; this 
class of questions was first treated by Mr. Spottiswoode in his paper “ On the Contact of 
Conics with Surfaces,” and general formulae applicable to all such questions were given. 
The results of that paper are here reproduced with some additions ; with the excep- 
tion of a few collateral theorems, these are all contained in the following Table : — 
^Number of five-point conics through fixed point = 6 
*Order of surface formed by five-point conics through fixed axis . = 8 
Number of six-point conics through fixed axis = 9f 
^Number of seven-point conics =70 
The second part treats of the contact of a quadric surface with a surface of the order 
n; and in particular it determines the number of points at which a quadric (other than 
the tangent plane reckoned twice) can have four-branch contact with the surface. In 
his paper “ On the Contact of Surfaces,” Mr. Spottiswoode proves that at an arbitrary 
point on a surface there is no other solution than the doubled tangent plane, and gives 
the conditions that must be satisfied by those points at which another solution is possible. 
The method here adopted is an extension of that applied by Joachimstal to the 
contact of lines with curves and surfaces. The coordinates of a point on a conic are 
expressed in terms of a single parameter, those of a point on a quadric by two para- 
meters. To determine the intersection with a given surface we have an equation in the 
parameter or parameters, and the conditions of contact are expressed in terms of the 
coefficients of that equation. The special case of the intersection of a quadric with a 
cubic surface is treated by the method of representation on a plane. 
* These results constitute the additions. 
t [Note by W. Spottiswoode. — In the Memoir quoted by Professor Ceiefobd, it was stated that the number 
of conics passing through a given axis and having six-pointic contact with a surface at a given point is ten. 
In making this statement I overlooked the fact that, in order to put in evidence that a certain quantity was a 
factor of the equation which determines the positions of the planes of the conics, the equation was multiplied 
by a quantity D which is a linear function of the position. In reckoning the degree of the equation this factor 
must of course he discarded. The degree is con sequently less by unity than that stated in the Memoir ; viz. 
it is 9, as proved by Professor Clifpobd. — July 3, 1873.] 
