SIR WILLIAM THOMSON ON VORTEX MOTION. 241 



if in its other terms the value of tan -1 ^ is reckoned continuously round from 

 one side of the plane ZOX to the other : or 



-»«#**(£)„,. 



if the continuity be from one side of ZOY to the other ; to render it really equal 

 to the first member of (1). Thus, taking for example the first form of the 

 added term, we now have for the corrected double equation (1) for the case of 



<p' = tan -1 - , (p any single valued function, and S the surface, composed of the 



two co-axal cylinders and two parallel planes specified above : 



dtp d<p 



fff X dz~ y dx dx dydz = = 2* ffdx dz (^) + ffdg tan" 1 h<p 

 JJJ x 2 + y 2 JJ \dyJy = Q JJ x 



dxdydz tan - l ^v 2 <p . . . (6). 



-Ill 



x 



But if we annex to S any barrier stopping circulation round the inner 

 cylindrical core, all ambiguity becomes impossible, and the double equa- 

 tion (1) holds. For instance, if the barrier be the portion of the plane ZOX, 

 intercepted between the co-axal cylinders and parallel planes constituting the 

 S of § 55, so that ffd<r must now include integration over each side of this 

 rectangular area; (6) becomes simply the strict application of (1) to the case 

 in question. 



57. The difficulty of the exceptional interpretation of Green's theorem for the 

 class of cases exemplified in §§ 55 and 56, depends on the fact that /Yds may have 

 different values when reckoned along the lengths of different curves, drawn within 

 the space bounded by S, from a point P to a point Q, ; ds being an infinitesimal 

 element of the curve, and F the rate of variation of <p per unit of length along it. 

 Let PCQ, PC'Q be two curves for which the fFds has different values ; and let 

 both lie wholly within S. If we draw any curve from P to Q ; make it first 

 coincide with PCQ, and then vary it gradually until it coincides with PC'Q ; it 

 must in some of its intermediate forms cut the bounding surface S : for we have 



Yds = -® dx + — =? dy + -$-dz 

 dx dy dz 



throughout the space contained within S, and -£ c> j-, -^ , are each of them 

 unambiguous by hypothesis ; which implies that f¥ds has equal values for all 



VOL. XXV. PART I. 3 Q 



