14 RICE ART. B 



the sum of the principal curvatures at a point x',y' z' on the 

 surface can be calculated as follows: Let p and q represent the 

 values of the differential coefficients bf/dx and df/dy when the 

 values x', y' are substituted for x, y, and let r, s, t be the values 

 of the second differential coefficients d^f/dx^, d^f/dxby, d^f/dy"^ 

 with the same substitutions; then 



, _ (1 + 9^) r + (1 + p^) ^ - 2 pqs 

 "'^''~ (1 + P^ + 3^)i 



This formula is used in obtaining equation [620] on page 283. 

 Its proof will be found in any text of analytical solid geometry. 



On page 293 of Gibbs, Vol. I, there is a reference to the total 

 curvatures of the sides of a plane curvilinear triangle. The 

 total curvature of an arc of a plane curve is equal to the external 

 angle between the tangents at its extremities and must be care- 

 fully distinguished from the average curvature of the arc which is 

 the quotient of its total curvature by its length. The angles of 

 the curviHnear triangle abc (Fig. 2) are YaZ, ZhX, XcY. Their 

 sum exceeds the sum of the angles of the plane triangle ahc by 

 Z Xbc-\- Z Xch -]- ZYca-\- Z Yac+ Z Zah + Z Z6a which is 

 equal to the sum of the external angles at X,Y, Z between 

 the tangents. This result is cited on page 293 of Gibbs, I. 



In conclusion it should be realised that Ci and C2 for a surface 

 may have different signs so that the expression Ci + d may 

 sometimes actually denote the numerical difference of the prin- 

 cipal curvatures of a surface at a point. This occurs when the 

 two principal sections produce curves which are convex to dif- 

 ferent parts. For example if one considers a mountain pass at 

 its top lying between hills on each side, a vertical section of the 

 surface of the mountain at the top of the pass made right across 

 the traveller's path is concave upwards, while one made at right 

 angles to this following the direction of traveller's path is con- 

 cave downwards. The principal centres of curvature are on 

 opposite sides of the surface in such a case and the principal 

 radii of curvature are directed to opposite parts. The radii 

 have opposite signs and the principal curvatures likewise. A 

 surface is said to be "anticlastic" at such a point (as opposed to 

 "synclastic," when the centres of curvatures are on one side and 



