1890.] On Stokes's Current Function. 47 



it is excluded from the region to which the expressions apply. The 

 position of the origin upon the axis is arbitrary, since by a trans- 

 ference formula we may pass from one origin to another. 



Let us now consider the system formed by a line source and a line 

 sink, of equal strengths, extending along the axis from an arbitrary 

 origin to infinity in opposite directions. Such a system I shall call 

 an extended doublet, of strength m, where m is the strength per unit 

 length of that part which lies on the positive side of the origin. 



By the superposition of two extended doublets, of equal but 

 opposite strengths, we ca,n produce a sink or a source upon the axis. 

 Hence, in a liquid, any irrotational motion which is symmetrical 

 with respect to an axis, may be produced by superposition of extended 

 doublets, whose origins depart but little from an arbitrary point on 

 the axis of symmetry. 



Now for an extended doublet of strength m, Stokes's current func- 

 tion ty, for any point distant r from the the origin, is 2mr. For let 

 be the distance of the origin of the doublet from the origin of co- 

 ordinates, and let ^(m, g ) be the value of Stokes's current function for 



any point (-sr, z). Then if Sty be the current function for a source of 

 strength 2mT> at the point of the axis, we get 



1 d 

 . o^ = 0. 



iff dr 



d 



