240 SIR WILLIAM THOMSON ON VORTEX MOTION. 



surface, S ; ffdrr integration over the area of this surface ; and t! rate of variation 

 per unit of length in the normal direction at any point of it. This is Green's 

 original theorem, with Helmholtz's limitation added (in italics.) The reader may 

 verify it for himself. 



56. But if either <p or cp' is a many-valued function, and the differential co- 

 efficients -?-> •■•> -£-, ■■■> each single- valued, the double equation (1) cannot 



be generally true. Its first member is essentially unambiguous ; but the process 

 of integration by which the second member or the third member is found, would 

 introduce ambiguity if <p or if <pf is many- valued. In one case the first member, 

 though not equal to the ambiguous second, would be equal to the third, provided 

 <p' is not also many- valued; and in the other, the first member, though not equal 

 to the third, would be equal to the second, provided <j> is not many- valued. 

 For example, let 



p' = tan- 1 1 , . . . (2). 



and let S consist of the portions of two planes perpendicular to OZ, intercepted 

 between two circular cylinders having OZ for axis, and the portions of these 

 cylinders intercepted between the two planes. The inner cylindrical boundary 

 excludes from the space bounded by S, the line OZ where <p' has an infinite 



number of values, and J- , and -7- have infinite values. We have 



' ax az 



dtp' — y dtp' x 



dx x 2 + y 2 ' dy ~~ x 2 + y 2 



(3), 



and at every point of S, d<p' = 0. Then, if </> be single-valued, there is no failure 

 in the process proving the equality between the first and second members of (1), 



which becomes 



dp _ d? 



dy dx> 



-4 o— dx dy dz = . . (4Y 



# + y 



Iff 



Compare § 14 (6) to end. 



The third member of (1) becomes 



I J dfftan- 1 ^ ^9-JJI tan_1 1 -v 2 <pdxdydz . . (5), 



which is no result of unambiguous integration of the first member through the 

 space enclosed by S, as we see by examining, in this case, the particular mean- 

 ing of each step of the ordinary process in rectangular co-ordinates for proving 

 Green's theorem. It is thus seen that we must add to (5) a term 



2*ffdxdz(*£) 



JJ \dyj y = 



