65 G6.] EXTENSION OF GREEN S THEOREM. 63 



Thomsons Extension of Greens Theorem. 



66. It was assumed in the proof of Green s Theorem that &amp;lt;f&amp;gt; 

 and T/T were both single-valued functions. If either be a cyclic 

 function, as may be the case when the region to which the inte 

 grations in (21) and (22) refer is multiply-connected, the statement 

 of the theorem must be modified. Let us suppose, for instance, 

 that &amp;lt;/&amp;gt; is cyclic ; the surface-integral on the left-hand side of (21), 

 and the second volume-integral on the right-hand side, are then 

 indeterminate, on account of the ambiguity in the value of &amp;lt;f&amp;gt; itself. 

 To remove this ambiguity, let the barriers necessary to reduce 

 the region to a simply-connected one be drawn, as explained in 

 Art. 54. We may suppose such values assigned to &amp;lt; that it shall 

 be continuous and single-valued throughout the region thus modi 

 fied (Art. 56) ; and equation (21) will then hold, provided the two 

 sides of each barrier be reckoned as part of the boundary of the 

 region, and therefore included in the surface -integral on the 

 left-hand side. Let da^ be an element of one of the barriers, /^ the 



cyclic constant corresponding to that barrier, -f the rate of varia- 



CLiv 



tion of A/T in the positive direction of the normal to dcr^ Since, 

 in the parts of the surface-integral due to the two sides of 



d&amp;lt;r v -f is to be taken with opposite signs, whilst the value of cf&amp;gt; 



on the negative side exceeds that on the positive side by K , we 

 get finally for the element of the integral due to da v the value 



~ KI On ^&quot; 1 -^- ence ( 21 ) b ecomes &amp;gt; i n tne altered circumstances, 



(30); 



where the surface-integrations indicated on the left-hand side 

 extend, the first over the original boundary of the region only, 



