646 



RICE 



ART. L 



ca, ah will be circular with their convexity outward if po > y^, 

 but with their convexity inward if po < Pa. In general 

 however Pa, Pb, Pc would not be equal, and in that case the lines 

 da, dh, dc with their continuations would be curved also, and 

 the convexity or concavity of any of the lines be, ca, ab would 

 be determined by the conditions as to whether Pd > Pa or po < 

 Pa, etc. If Pd = Pa; of course be is straight. (To avoid 

 awkward digression later we deal with a few geometrical facts 



Fig. 7 



now. The total eurvature of a limited curved line is the exterior 

 angle between the tangents at its extreme points and is equal 

 to the sum of the two angles between the chord joining these 

 points and the tangents. The angles of a curvilinear triangle 

 are the angles between the pairs of tangents drawn to pairs of 

 adjacent sides where they meet. It will be easily seen that the 

 excess of the sum of the angles of a curvilinear triangle over two 

 right angles is equal to the algebraie sum of the total curvatures 



