It seems desirable that the methods taught to beginners of reducing the general equation of the second degree to the standard forms should be based on elementary properties of conics directly deduced from the standard equations. The methods given below are suggested as being simpler than those indicated in textbooks or in the references quoted below. It is presumed that pupils are already conversant with the forms and names of the conics represented by the equations (x/p)2±(y/q)2= 1, y2 = 4nx, of the existence of the centre and axes of symmetry (principal axes) of the central conics, and of the axis and the tangent at the vertex of the parabola. No other property of the conic is assumed.

While the conception of the Circle at Infinity as a weapon for analytical investigation has been fairly often applied (as for example “Some equations connected with the Plane Section of the Conicoid” by the author, Math. Gaz., July 1933), its use as the basis for descriptive enquiry has been rarer and disconnected. The aim of this paper is to obtain by simple steps the principal descriptive properties of the quadric surfaces from their intersections with the Plane and Circle at Infinity. The method also has the advantage of making manifest how the basic causes of the differences between the properties of different types of quadric arise, which are often hidden in algebraic treatment.

Accessible accounts of confocal coordinates are somewhat diffuse, and the following synopsis may be helpful. It will be understood that my sole concern here is with the efficient handling of these coordinates; I am not denying that confocal conicoids must be seen as a linear tangential system if they are to be appreciated, or that pure geometry throws a brilliant light on their properties.