10 RICE ART. B 



minimum condition and the consideration of the problem goes 

 beyond the Hmits of possible discussion here. 



For the proof of these results see any text of modern algebra, 

 for example Bocher's Introduction to Higher Algebra, Chapters 

 IX-XII. 



For reference to these conditions in the Collected Works, 

 see Gibbs, I, pp. 111,112,242. 



2. Curvature of Surfaces. The average curvature of a plane 

 curve between two points A and B is defined as the quotient of 

 the external angle between the tangents at A and B by the length 

 of the arc AB. From a kinematic point of view it is the average 

 rate of rotation of the tangent per unit length travelled by the 

 point of contact. If the point B approaches indefinitely near 

 to A, the limiting value of the average curvature is defined to be 

 the curvature at the point A. In the case of a circle this is 

 obviously the reciprocal of the radius at all points. For any 

 curve at any point the curvature has the dimension of a recipro- 

 cal length, and so, on dividing the value of the curvature at a 

 point on a curve into unity, we obtain a definite length which is 

 then referred to as the "radius of curvature" at that point. 

 Clearly where the curvature is relatively large the radius of 

 curvature is relatively small; thus the extremities of the major 

 axis of an ellipse are the points on it at which curvature is great- 

 est but radius of curvature least ; at the extremities of the minor 

 axis, curvature is least, radius of curvature greatest. 



The measurement of curvature at a point on a surface is based 

 on this simple idea for a curve. Thus we conceive the tangent 

 plane and the normal line to be drawn at a point P on the sur- 

 face, and we then consider any line through P lying in this plane. 

 An infinite number of planes can be drawn cutting the tangent 

 plane in this hue. These planes will cut the surface in an in- 

 finite number of curves, and we w'ill readily recognise that suffi- 

 cient information concerning the curvature of these curves at 

 the point P will give us all the vital information concerning the 

 curvature of the surface at P. Two obvious details in the con- 

 struction of one such curve can be varied at will; we can alter 

 the angle between the tangent plane at P and the plane drawn 

 through the line in the tangent plane (the tangent line as we 



