OF ARTS AND SCIENCES. 21 



the origin, the line integrals of U cos 8 and CTsin S, taken around T, 



are equal respectively to the surface integrals of —^ and , 



taken over the area enclosed by T. 



For the element of plane surface in polar coordinates, rAr^O may 

 be used. Let the radius vector OP, drawn so as to make the angle 6 

 with the initial line OX, cut T 2n times at points P-^, P^, P3, .... P^n-, 

 distant respectively r^ , i\, r^, .... r^^ from 0. Let the values of U 

 at these points of intersection be Ui, U^, V^, .... f^jn? respectively. 

 Whenever the radius vector cm^s into the closed contour, either +S 

 or — 8 is an obtuse angle and cos 8 is negative ; whenever the radius 

 vector emerges from the space enclosed by the contour, either + S or 

 — 8 is acute and cos 8 positive. The two neighboring radii vectores, 

 OP and 0P\ which make with each other the angle A^, include be- 

 tween them the arcs As^, As2, ^Ss? A«4, .... Asjn, cut out of T, and 

 the arcs r^AQ, r^AQ , ^3^^, .... r2„A^, cut out of a set of circum- 

 ferences drawn about as centre, with radii r^, r^, r^, r^, .... r^n, 

 respectively. It is evident that, if A 6 be made to approach zero as a 

 limit, 



= -1. 



If the double integral be extended all over the space enclosed by T, 



fS^^^ '''^''^^ "" P^ ^~ '''^' + r,U,-r,Us + .... + r,,U,,-], 



where the integration with respect to 6 is to be extended over all 

 values of the angle for which the corresponding radii vectores cut T. 

 If now for r-iAO, r^AO, r^AO, etc., — cos Si ■ ds^, + COS82 -ds^, 

 — cos S3 • dss + ....+ cos San <^^2n t>6 Substituted respectively, we have 



I I ^J[S^ — -rdrd9= / [C^i cosSi£?Si + C^cos82C?S2+ •— ^nCos82„], 



and this last integral is evidently equal to the line integral of UcosS 

 taken all around T. 



It is to be noticed that, if Were within T, each radius vector would 

 cut T an odd number of times, and that a negative sign must stand 

 before the line integral. * 



