CURKKNTS IN C( )N |)I< I IMi SHKLLS Of SMALL THICKNESS. 299 



constant value along each stream line. Therefore there exists a function, <f>, which is 

 constant along each stream line, and such that d^dc is equal to the given superficial 

 current at every point, dc being an element of a line drawn on the surface at right 

 angles to the stream line. Then < is the required function. 



For U, V, and W so defined are proportional to the direction cosines of the common 

 section of the surface S, and the surface $ = constant, that is, to the direction cosines 

 of the stream line. 



Also 



V + W = (. + ) + (P + n 



, dd> d<b ,, dd> d<b 

 2ln r 



, 

 </: ax ay 



^ d<f> 



/d<t>\* 



U) - 



(*.. MM 



' \d v ') ~\d v ) 



(if d<f>/dv f be the rate ot increase of <f> per unit of length of the normal to the surface 

 <ft = constant, and d<f>/dv be the rate of increase of <j> per unit of length of the normal 

 to the sheet) 



fr 



There may be an infinite variety of functions which satisfy the conditions for <f>, 

 but all of them give the same value for U, V, and W, If ^ be given, it completely 

 determines U, V, and W. Conversely, if U, V, and W be given at every point, they 

 completely determine the values of <f> on S subject to the addition of an arbitrary 

 constant. 



Of the Currents per Unit of Area. 



4. Let there be any finite space, and two functions S and $ such that within the 

 space 



dS d<f> dS d<ft 

 dz dy dy dz 



JO J-t JQ J 



> ii<p rto u<p 



v = j~ j T ~r 



dx dz d* dx 



JC3 11 JQ 1 



</h ntp rto n<b 

 dy dx rfjf dy 



and hi + mv -|- mo = at all points on the bounding surface. 



2 Q 2 



