SECTION III. 1882. LS 18e] TRANS. Roy. Soc. CANADA. 
Symmetrical Investigation of the Curvature of Surfaces, ete. 
By ALEXANDER JOHNSON, M.A., LL.D., Dublin; Professor of Mathematics and Natural 
Philosophy, McGill University, Montreal. 
(Read May 25 1882.) 
The object of the following paper is to show that the leading propositions concerning 
the curvature of surfaces, may be obtained in a direct and simple manner by a symmetrical 
investigation, in which each proposition leads naturally to the following. 
The first step to this, is a simplified solution of the well known problem, “To find 
the equation, referred to its axes, of the plane section of a central quadric.” The paper 
may be considered as consisting of two parts: the first, referring to the axes of conics and 
quadrics ; and the second, to the curvature of surfaces specially. 
The following is a summary of these parts : 
I. AXES OF CONICS AND QUADRICS. 
1°. Symmetrical investigation of the magnitudes and directions of the axes of a plane 
section of a central quadric. 
2°. Geometrical interpretation of the analytical conditions. 
3°. Symmetrical solution of four homogeneous equations which give a symmetrical 
determinant. 
4°. Conditions that the section of the quadric be circular. 
5°. Application of same method :— 
(a) To find magnitude and direction of the axes of a quadric. 
(6) To find magnitude and direction of the axes of a conic. 
(c) To ‘the discussion of the nature of the plane sections of any quadric given by the 
general equation. 
II. CURVATURE OF SURFACES. 
6°. Investigation of the radius of curvature, at a given point of any surface, of any 
plane section through the point. 
7°. Deduction of the value of the radius of curvature of a normal section, and Meu- 
nier’s Theorem. 
8°. Equation, for a given point of any surface, of a quadric such that the squares of 
the semi-diameters of the section of it made by the tangent plane to the surface give the 
radii of curvature of the corresponding sections of the surface. 
9°. Value of the principal radii of curvature, and Euler’s formule. 
10°. Directions of maximum and minimum curyature. 
11°. Conditions for umbilics. 
12°. Lines of curvature. 
