BOCHER. — HARMONIC FUNCTIONS IN TWO DIMENSIONS. 583 



tinuity of these derivatives, recourse is ordinarily had to the theory of 

 functions of a complex variable. Instead of this, we may first deduce 

 from the last written differential equations the relation 



Qu 9v 

 9c 9t 



where o- is any direction and t the direction which makes a positive right 

 angle with cr. From this we infer that 



C9u. C9v , C9v , P9u, 



I =-ds=z I =-08, I =- ds = — / -tr- ds. 



J 9n J ds J dn J ds 



These integrals will therefore vanish When extended around a closed 

 curve, since u and v are both single valued. From this it follows that 

 U and v are both harmonic. 



Harvard University, 

 Cambridge, Mass. 



