1884.] which is moving Rotationally and part Irrotationally. 281 

 Calculate the quantity P = -—( ®lk +?2L 



Treating p as the density of a material distribution inside \=0, taking 

 no account of the value of p outside the surface \=0, obtain the 

 potential of this distribution. Let the potential inside \=0 be x' 5 and 

 outside let it be 0, 



X will, in general, differ from x j first, because x niay contain many- 

 valued terms, which may be denoted by P satisfying Laplace's equa- 

 tion; and, secondly, because x~ P m&y De the potential of a distribu- 

 tion of matter, part of which is outside X=0. 



Accordingly, it is necessary to examine whether it is possible to 

 find many- valued terms P satisfying Laplace's equation such that 



Then + P will be the velocity potential of the irrotational motion, 

 provided that it give zero velocity at infinity. 



The few illustrations which follow are a first attempt to apply the 

 above theory to particular cases. 



Example 1. y being rectangular co-ordinates in a plane, 



T any function of the time t, Y its differential co- 

 efficient with regard to the time. 

 /, a, b, constants, 



u, v the components of the velocity parallel to the axes, 

 it is shown that u=(f) ^~- S ; — Y— (/)?^ satisfy the equa- 

 tions of motion. 



The surfaces containing the same particles are the elliptic cylinders 



(/) (5 + -p^ 2 ) =constant- 



For a finite portion of the plane of as, y outside the cylinder 

 + ^ — Z2_ = l, the current function of an irrotational motion 



continuous with that inside the cylinder is- 



] + l 

 ab 



(/) —I"- log C^ + e+^ + e) 



w 5 a w r )( , t «)-.t 



where e, v are the elliptic co-ordinates satisfying the equations 

 and — 6 2 <e<oo , — a?<v< — b 2 , b<a. 



