20 



PROCEEDINGS OP THE AMERICAN ACADEMY 



III. 



ON SOME THEOREMS WHICH CONNECT TOGETHER 

 CERTAIN LINE AND SURFACE INTEGRALS. 



By B. O. Peirce. 



Presented May 13, 1891. 



In transforminw from one set of curvilinear coordinates to another, 

 some of the differential expressions which appear in problems in 

 Hydrokinematics and Electro kinematics, I have found the theorems * 

 stated below useful. 



Theorem. — Let U be any function of the two polar coordinates, 

 T and d, which, with its first space derivatives, is finite, continuous, 



Fig. 1. 



and single-valued throughout that part of the coordinate plane which 

 is shut in by the closed curve T. Let 8 be the angle between the ra- 

 dius vector, drawn from the origin to any point P on T, and the normal 

 to T drawn from within outwards at P. Then, if 7' does not include 



* London Educational Times, January and February, 1891. 



