PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 75 



Now, any finite portion of a surface may be broken up into small elements such 

 as we have just treated, and the sign only of the integral along each portion of 

 a bounding curve is changed when we go round it in the opposite direction. 

 Hence, just as Ampeee did with electric currents, substituting for a finite closed 

 circuit a network of an infinite number of infinitely small ones, in each con- 

 tiguous pair of which the common boundary is described by equal currents in 

 opposite directions, we have for a finite unclosed surface 



fS.adp =jrS.V<rUv.ds. 



There is no difficulty in extending this result to cases in which the bounding 

 curve consists of detached ovals, or possesses multiple points. This theorem 

 seems to have been first given by Thomson (Thomson & Tait's " Natural 

 Philosophy," § 190 (j) ; Thomson on Vortex Motion, Trans. R.S.E., 1868-9, 

 § 60 (?) )> where it has the form 



/(«* + ** + ■& =Jr* ('(I -f ) + -(£-£) + Ki-t)) • 



It solves the problem suggested by the result of § 8 above. 



12. If o- represent the vector force acting on a particle of matter at 



p, — S.crdp represents the work done while the particle is displaced along dp, 



so that the single integral 



J* S.crdp 



of last section, taken with a negative sign, represents the work clone during a 

 complete cycle. When this integral vanishes it is evident that, if the path be 

 divided into any two parts, the work spent during the particle's motion through 

 one part is equal to that gained in the other. Hence the system of forces must 

 be conservative, i.e., must do the same amount of work for all paths having the 

 same extremities. 



But the equivalent double integral must also vanish. Hence a conservative 

 system is such that 



ffds&.VcrUv= 0, 



whatever be the form of the finite portion of surface of which ds is an element. 



Hence, as Vcr has a fixed value at each point of space, while Uv may be altered 



at will, we must have 



VVo- = 0, 

 or 



Vo- = scalar. 



If we call X, Y, Z the component forces parallel to rectangular axes, this 

 extremely simple equation is equivalent to the well-known conditions 



^X_^Y_ dY __dZ_Q ^_^ = o 

 dy dx ' dz dy ' dx dz 



