OF ARTS AND SCIENCES. 23 



ds^ = ~, and for the element of surface -— — ^ may be used. The 



"l / y\ "1 ' "2 



surface integral of hi-Ii.^.D^ i- J taken over the area enclosed by 



7' is 



o =JJh ■ K ■ A (;|-^) ds, . ds, =Jd r/Jn^ (J^ dc 



Consider two curves Q, 0' Q' along which r/ has respectively the 

 constant values r]o and r/o + A r/ ; and let ^ increase in the directions 

 OQ, O'Q'. 



Let OQcutT 2n times at the points P', P", P'", ....,P^'-"\ where 

 the values oi' /u are h^', h.^", h.^", .... , /^j'""', respectively, and the cor- 

 responding values of V, V, V", V", ...., f^^-"\ The curved line 

 Q makes with the normals drawn to T at P', P", P'", etc., from 

 within outwards the angles 8', 8", l'", etc., and the two curves OQ, 

 O'Q', cut out of 7" the 2» arcs As', As", As'", ...., AP"K 



where the integration is to be extended over all values of rj which 

 occur within T. 



The angles 5', 5'" 5^^"""'^, or their negatives, are all obtuse and 



their cosines are negative, but the angles 8", 8^'^', .... S^-"\ or their 

 negatives, are all acute and their cosines are positive, so that at every 

 point, P^"', where OQ cuts 7' we have 



Limit r (-!)-• A s^-^ cos 8r-n _^ 



d n 



and in the expression for O we may write ( — 1)'^ • cos S™ c?s™ for yj^.. 



"2 



Hence, 



where the sign of integration directs us to find a similar expression to 

 that in the brackets for every pair of consecutive curves of constant r/ 

 which cut T, and to find the limit of the sum of the whole. This is 

 evidently equivalent to integrating T^cosS all around the curve T. 



Jefferson Physical Laboratory, 

 Cambridge. 



