CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 457 



The product of F (w) and its conjugate surd is a quadratic expression obviously 

 possessed of real zeros, and so the only zeros for which F (iv) need be tested are real 

 ones. If the case of k > c be first considered, it is seen that F(- oo), F(-c), F(c) 

 are negative, while F (k) and F ( + co ) are positive. Hence F (w) has just one zero, 

 namely, between c and k, the corresponding value g of w being 



= [X%-{x*P-(x s -l)cT'']/(\ s -l); ...... (53) 



the quantity g' derived from this by changing the sign of the square root is a zero 

 of the expression conjugate to F (?/;). 



If k<-c, say k = -k', it is seen that F(- co) and F(-&') are negative, while 

 F(-c). F(c), and F(+oo) are positive. Thus F(?r) has just one zero, namely, 

 between k' and c, and its value h is given by 



\W-(\'-l)W]/(\>-l); ..... (54) 



the quantity h' derived from this by changing the sign of the .square root is a zero 

 of the expression conjugate to F (ir). 



In each of these cases the zero of F(?) is of the first order. Hence if in the 

 alternative cases attention be directed to (w}/(ivg) and F(w)f(iv+Ji) respectively, 

 it is seen that these are functions free from zeros and infinities in the relevant region. 

 Thus for k > c, k' > c, and X > 0, 



(55) 



w + h 



are conformal curve-factors. In each case the linear range is c to c, the angular 

 range is zero, and the order at infinity is zero. Within the linear range the modulus 

 of is v/{(X 2 -l) (g'-u')/(g-ic)} and that of \, is v/{(X 2 -l) (- + l/)/(u- + !>)}. It 

 is to be noticed that, if A > 1, g' > k and h' > k f ; if\<l,g'<c and // < c. 



Any power of *$ u or ^ 12 is a curve-factor of inflexional character, and the greater 

 the power the more pronounced is the variation in the direction of the tangent at 

 different points of the curve. ^ n corresponds to a curve which, as w decreases from 

 c to c, is first concave and afterwards convex to the relevant region ; ^ corresponds 

 to a curve which is first convex and afterwards concave. But any negative power of 

 either corresponds to a curve in which, as compared with the original, the order of 

 convexity and concavity is reversed. 



22. Hydrodynamical Example. The specification of a liquid stream disturbed by 



3 P 2 



