184] 



TORSION OF PRISMS. 



487 



If we now form the line - integral around the contour of the cross- 



dV 

 section, of -? from 109) , 



on 



dV 



ll7 ) /fe rfs =/(E cos (^)+ 



= / [ y cos (nx) -\- x cos 



7 

 ds 



cos (nx) cos (nyy\ ds, 



i a y v '-v w "V $ x 

 since if we circulate anti- clockwise (Fig. 152), 



ds cos (w#) = dy, ds cos (ny) = dx 

 the integrals become 



118) 



= - / (ydy + xdx), 



*J 



and since both differentials are perfect, 

 integrating, 



119) gr^C-ite'+tf 2 ). 



Fig. 152. 



Accordingly if we can determine a harmonic function *P which on 

 the contour shall have the value 119), the problem is solved. For 

 example take the functions, 



giving 



equal to a constant on the ellipse 



tf z , y' 



a 2 -t- -ftl 



if we take 



giving 



~ 2 



