458 DK, J. G. LEATHEM ON SOME APPLICATIONS OF 



the presence of a pear-shaped body of the general character shown in fig. 6 may be 

 got by combining a Schwarzian factor, a curve-factor of angular range %TT for the 



Fig. 6. 



forward curve, and a power of ^ 12 for the inflexional curve. Such a transformation 

 would be 



, U (-6) + {(w-a)(w-c)Yiy* C dw , . 



vCu+iyMx+iy^-a)''* 



where a > l> > c. The condition for the second straight stream-line being in the 



production of the first is 



V 



/u. + 



CURVE-FACTORS OF SEMI-INFINITE LINEAR RANGE. 



23. For a range extending from w = to w oo the simplest type of curve- 

 factor is 



= .'+,* ......... (59) 



where a''- > 0. Its angular range is \ir and its order at infinity is |-. The trans- 

 formation 



gives a boundary consisting of a straight line and a curve intersecting it at an angle 

 p-n-. In the special case of p = % the curve is a semi-parabola. 



It may "be noted that when w is negative | ^ 13 | = (awf ! \ 



24. If a more general expression be considered, namely 



V = w-k+ (cw) 11 *, ...... ... (61) 



where c > 0, it is found that a distinction has to be drawn between the case when 

 Jc > and the case when k < 0. 



