60 IRROTATIONAL MOTION. [CHAP. Ill 



Lord Kelvin's Extension of Greens Theorem. 



53. It was assumed in the proof of Green's Theorem that 

 and <' 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 Art. 43 refer is multiply-connected, the statement of the 

 theorem must be modified. Let us suppose, for instance, that <f> 

 is cyclic ; the surface-integral on the left-hand side of Art. 43 (5), 

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

 indeterminate, on account of the indeterminateness in the value of 

 </> itself. To remove this indeterminateness, let the barriers neces- 

 sary to reduce the region to a simply-connected one be drawn, as 

 explained in Art. 48. We may now suppose (f> to. be continuous 

 and single-valued throughout the region thus modified; and the 

 equation referred to 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 ScTj be an element of one of the barriers, K^ the cyclic constant 

 corresponding to that barrier, dfi/dn the rate of variation of </>' in 

 the positive direction of the normal to So-!. Since, in the parts 

 of the surface-integral due to the two sides of 8<r lt dfi/dn is to be 

 taken with opposite signs, whilst the value of < on the positive 

 side exceeds that on the negative side by K I} we get finally for the 

 element of the integral due to 8^, the value ^dfi/dn.^. 

 Hence Art. 43 (5) becomes, in the altered circumstances, 



t AT, + *, d<r, + &c. 



dn JJ dn 



+ + dxd dz 



dx dx dy dy dz dz 



(1); 



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

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

 and the rest over the several barriers. The coefficient of any K is 

 evidently minus the total flux across the corresponding barrier, 

 in a motion of which </>' is the velocity-potential. The values of $ 

 in the first and last terms of the equation are to be assigned in 

 the manner indicated in Art. 50. 



