79-81] FLOW IN A CURVED STRATUM. 115 



In the case of uniform thickness, to which we now proceed, it 

 is convenient to write ^r for ty/h, so that the velocity perpendicular 

 to an element s is now given indifferently by d^rjds and d<f>/dn, 

 8n being an element drawn at right angles to Ss in the proper 

 direction. The further relations are then exactly as in the plane 

 problem ; in particular the curves <f> = const., -fy = const., drawn for 

 a series of values in arithmetic progression, the common difference 

 being infinitely small and the same in each case, will divide the 

 surface into elementary squares. For, by the orthogonal property, 

 the elementary spaces in question are rectangles, and if 8s l} Ss 2 be 

 elements of a stream-line and an equipotential line, respectively, 

 forming the sides of one of these rectangles, we have dyfr/ds 2 

 = d$/ds l} whence Bs l = Ss 2 , since by construction &*/r=^>. 



Any problem of irrotational motion in a curved stratum (of 

 uniform thickness) is therefore reduced by orthomorphic projection 

 to the corresponding problem in piano. Thus for a spherical 

 surface we may use, among an infinity of other methods, that 

 of stereograph ic projection. As a simple example of this, we may 

 take the case of a stratum of uniform depth covering the surface of 

 a sphere with the exception of two circular islands (which may be 

 of any size and in any relative position). It is evident that the 

 only (two-dimensional) irrotational motion which can take place 

 in the doubly-connected space occupied by the fluid is one in 

 which the fluid circulates in opposite directions round the two 

 islands, the cyclic constant being the same in each case. Since 

 circles project into circles, the plane problem is that solved in 

 Art. 64, 2, viz. the stream-lines are a system of coaxal circles with 

 real 'limiting points' (A, B, say), and the equipotential lines are 

 the orthogonal system passing through A, B. Returning to the 

 sphere, it follows from well-known theorems of stereographic pro- 

 jection that the stream-lines (including the contours of the two 

 islands) are the circles in which the surface is cut by a system of 

 planes passing through a fixed line, viz. the intersection of the 

 tangent planes at the points corresponding to A and B, whilst 

 the equipotential lines are the circles in which the sphere is cut 

 by planes passing through these points*. 



* This example is given by Kirchhoff, in the electrical interpretation, the 

 problem considered being the distribution of current in a uniform spherical 

 conducting sheet, the electrodes being situate at any two points A, B of the surface. 



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