524 XL HYDRODYNAMICS. 



components u and v, let us call the two conjugate functions (p and ^ 7 

 satisfying the equations 



It is evident that qp and ^ have the relation of the velocity potential 

 and stream function denned in 195. If one function is given we 

 can find the conjugate, for we must have 



which by equations 122) is 



Now if we call this Xdx + Ydy it satisfies the condition for a 

 perfect differential 



dX __ dY 

 dy dx 

 that is, in this case, 



dy* dx* 

 Consequently the line integral 



dtp -. . dm -. } 



dx + 7T 2 - dy \ 



dy dx y ) 



from a given point # , y Q to a variable point x, y, is a function only 

 of its upper limit and represents ^. Similarly if ^ is given 



123) , 



Furthermore the first of the equations 122) is the condition that 

 V dx + <JP dy is a perfect differential and the second that 

 is such.. Accordingly the line integrals 



= I \tydx -f <pdy}, 



124) J 



give two new point functions 3> ? ^F which in virtue of the equations 



d$ dW 



ib = 75 = -- o 9 

 dx dy 



125 ) = a* = ^ 



^ 3y dx' 



are conjugate to each other and give a new analytic function of z, 



