196 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



The function p represents the flux for any tube bounded by 

 the surfaces /u- = 0, //, = const, and two planes through the ^-axis 

 making a dihedral angle with each other equal to unity, and ^ is 

 then called Stokes's current- or flux-function. 



104. Functions of Complex Variable on Surface. Both 

 of the cases just considered are cases of a class of problems of 

 considerable generality. If the vector lines lie in one of the 

 coordinate surfaces itself, we have the particular integral 

 X = ft = const., and accordingly 



ff h, 3F, h, dV, } 



(1) //, = MT-T- 5 a<7i nr o~ "ftr = const. 



}\h&dq t h z h 3 dq l ^} 



or the differential 



/ \ j ty j 9/A 7 h 2 dV .. ^ dV , 



(2) dfji ~ dq l + ~- rfft = T-T- r dq 1 - ,- T - ^ dq t . 



0$i dq 2 hj^ dq 2 h 2 h 3 dq^_ 



From this we must have 

 dj^ 



Differentiating the first of these equations by q 2 , the second by 

 , and adding, 



_ 



9ft ^, 9ft J 



Expressing the derivatives of V in terms of those of /i, 

 9F = M 1 9^ _9F = ^A.9/i 



8ft ~~ A 2 8ft ' 8ft ~ A! 8ft ' 



Differentiating the first by ft, the second by ft, and adding, 



A f^. 1 ?^l j_ J i - n 



3ft U aftraftt^ 8ftj~ 



Now if ^ 3 is independent of ft and ft, which will be the case 

 if two consecutive surfaces ft, ft + dft are parallel, or everywhere 

 the same distance apart, namely, 



then 3 comes out as a factor of both differential equations, and 

 we find that V and p satisfy the same differential equation 



