99 



PROCEEDINGS OF THE AMERICAN ACADEMY 



Since the limit of the ratio of Ar to any one, QQ', of the arcs cut 

 out of T by two circumferences of radii r and r -{■ \r respectively, 

 drawn around (? as a centre, is equal in absolute value to the sine of 

 the angle w^iich Q makes with the external normal to T at Q, it is 

 easy to prove the second part of the theorem by integrating D^ U with 

 regard to 6 first, and then, after introducing proper limits, with regard 

 to r. 



This theorem may be regarded as a useful special case of the 

 following 



Theorem. — Let I =z f^ (x, y) and -q =: f^ {x, y) be two analytical 



functions of a: and y such that the two families of curves fi (x, y) = c, 



f^ix, y) = Z:, are orthogonal. Let Vhe any function of x and y which, 



with its first space derivatives is finite, continuous, and single-valued 



within a closed curve T, drawn iu the coordinate plane. Let hi and 



Fig. 2. 



^2 be the positive roots of the equations Iti^ = (D^ tf + {Dy ^)^, 

 h^ — (Z)^ r;)- + (Dyrj)'^. Then, if ^ has neither maximum nor mini- 

 mum values within 7] the surftice integral of /«i • //o • />,; I — ), taken 



all over the area enclosed by T is equal to the line integral taken 

 around T'of FcosS, where 8 is the angle between the exterior normal 

 drawn to T at any point, and the curve of constant 77 drawn through the 

 point, and where the direction in which ^ increases is taken positive. 

 Similarly, if proper regard be had for signs, 



f p'l ■ h A {T) ^« = r^sin 8 . ds. 



If through any point, P, in the coiirdinate plane, two arcs .v, , So be 



drawn along which ^ and ij are respectively constaut, d Sy = '' , 



2 



