93-94] SYMMETRY ABOUT AN AXIS: STREAM-FUNCTION. 135 



94. The velocity-potential due to a unit source at the origin 

 is 



* = l/r .............................. (1). 



The flux through any closed curve is in this case numerically equal 

 to the solid angle which the curve subtends at the origin. Hence 

 for a circle with Ox as axis, whose radius subtends an angle 6 at 0, 

 we have, attending to the sign, 



2-m/r = - 2-7T (1 - cos 6). 

 Omitting the constant term we have 



x dr . 



dr = -=-=- .............................. (2). 



r dx 



The solutions corresponding to any number of simple sources 

 situate at various points of the axis of x may evidently be super- 

 posed ; thus for the double-source 



d 1 cos 6 



d z r OT 2 sin 2 

 we have ^ = - = -- - = --- ............... (4). 



dx 2 r 3 r 



And, generally, to the zonal solid harmonic of degree -n 1, 

 viz. to 



corresponds -^ = A (6)*. 



CvU/ 



A more general formula, applicable to harmonics of any 

 degree, fractional or not, may be obtained as follows. Using 

 spherical polar coordinates r, 6, the component velocities along 

 r, and perpendicular to r in the plane of the meridian, are 

 found by making the linear element PP' of Art. 93 coincide 

 successively with rW and Sr, respectively, viz. they are 



1 d* Id* ^ 



r sin 6 rdQ ' r sin 6 dr " 



* Stefan, " Ueber die Eraftlinien eines um eine Axe symmetrischen Feldes," 

 Wied. Ann., t. xvii. (1882). 



