458 Mr MAXWELL, ON THE 



Then the value of u will determine the position of a curve of the first system, and 

 that of u a curve of the second system, and therefore u and u will suffice to determine the 

 point of intersection of these two curves. 



Hence we may conceive the position of any point on the surface to be determined by the 

 values of u and u for the curves of the two systems which intersect at that point. 



By taking into account the equation to the surface, we may suppose x, y, and z the 

 co-ordinates of any point, to be determined as functions of the two variables u and u. This 

 being done, we shall have materials for calculating everything connected with the surface, 

 and its lines of bending. But before entering on such calculations let us examine the 

 principal properties of these lines which we must take into account. 



Suppose a series of values to be given to u and u, and the corresponding curves to be 

 drawn on the surface. 



The surface will then be covered with a system of quadrilaterals, the size of which may 

 be diminished indefinitely by interpolating values of u and u between those already assumed ; 

 and in the limit each quadrilateral may be regarded as a parallelogram coinciding with a facet 

 of the inscribed polyhedron. 



The length, the breadth, and the angle of these parallelograms will vary at different parts 

 of the surface, and will therefore depend on the values of u and u . 



The curvature of a line drawn on a surface may be investigated by considering the 

 curvature of two other lines depending on it. 



The first is the projection of the line on a tangent plane to the surface at a given 

 point in the line. The curvature of the projection at the point of contact may be called 

 the tangential curvature of the line on the surface. It has also been called the geodesic 

 curvature, because it is the measure of its deviation from a geodesic or shortest line on the 

 surface. 



The other projection necessary to define the curvature of a line on the surface is on a 

 plane passing through the tangent to the curve and the normal to the surface at the point 

 of contact. The curvature of this projection at that point may be called the normal 

 curvature of the line on the surface. 



It is easy to show that this normal curvature is the same as the curvature of a normal 

 section of the surface passing through a tangent to the curve at the same point. 



10. General considerations applicable to the inscribed polyhedron. 



When two series of lines of bending belonging to the first and second systems have 

 been described on the surface, we may proceed, as in Art. (l), to describe a third series of 

 curves so as to pass through all their intersections and form the diagonals of the quadrilaterals 

 formed by the first pair of systems. 



Plane triangles may then be constituted within the surface, having these points of 

 intersection for angles, and the size of the facets of this polyhedron may be diminished 

 indefinitely by increasing the number of curves in each series. 



But by Art. (8) the first and second systems of lines of bending are conjugate to each 

 other, and therefore by Art. (4) the polygon just constructed will have every pair of triangular 



