CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 485 



: IT = pir : ( 1 p) TT, so that m=p/(lp); similarly n q/(lq). Thus the 



transformation is 



p _2_ 



a 7'. c /z- ^y/l'r dz \* = c **** ( i34 ^ 



with the conditions 



c > a t > a 2 > a 3 ... > c, (135) 



and 



.... (136) 



the latter being the condition that the transformation have zero angular range. In 

 the application to ships there must also be the condition for vanishing of the source 

 term in the expansion of w for z great. 



Clearly it is possible, with obvious precautions, to introduce on the left-hand side 

 of (134) curve-factors in z having a linear range inside the range c to c, due 

 account being taken of their angular range in (136). 



A particular example (as regards the w configuration rather a limit case) of the 

 transformation (134) is got by putting/) = f, q = |-, KI = 1, /r a = 1, i = ft, 2 = . 

 This gives 



Gdw=~~dz, (137) 



Z (Jb 



wherein c > a. The curves in the z plane corresponding to this transformation are 

 KANKINE'S " oval neoids."* 



51. The transformation for the semi-ellipse (c cosh a, c sinh a) and the productions 

 of its minor axis, written in the form 



(z 2 c 2 sinh 2 a)dz 7 /ir, \ 



aWl , r- / 2 a w i . = dw > ( 138 ) 



c ) " {z cosh a+ (z +c ) " sum a} 



exemplifies the possible presence in the denominator on the left-hand side of (i) a 

 function having imaginary zeros which are not in the relevant region, (ii) a function 

 entirely free from zeros, and possessed of branch points outside the relevant region. 

 Doubtless it would be possible to effect some generalisation of a formula including 

 these features. 



SUPPLEMENTARY NOTE ON CURVE-FACTORS. 



* 



(Added 15th June, 1915.) 



52. It should be noticed that the product of two curve-factors, of which the linear 



range of one is contained in the linear range of the other, is itself a curve-factor for 



the greater linear range. The resultant curve-factor is not, however, simple in the 



sense specified in 4, for though a curve represented by the curve-factor in any 



* RANKINE, " On Plane Water-Lines in Two Dimensions," ' Scientific Papers,' p. 495. 



