CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 4H7 



The undulations which characterise ^ 44 do not appear in curves represented by W^,. 

 The relation corresponding in this case to formula (139) is 



(141) 



.ft 



An example of ^ 15 , got by putting f(0) equal to a constant y. is 



ll - u - M ..... (142) 



subject to 



-b] = 1 ......... (143) 



54. A generalisation of *$ ^ is 



# 17 = (*/-*,)"'( -*..)-... +(w-a)'(w-c 1 ) ni (w-c. t )^...(iv-b) 9 , . (144) 



where the k's lie between a and b but do not coincide with any of the c's, the //<'s are 

 all positive, and 



It is readily verified that, for real values of w between n and b, the imaginary part 

 of ^. 17 does not vanish, while the real part vanishes an odd number of times. Hence 

 the angular amplitude is TT. 



Treatment- on the lines of article 53 leads to the form 



48 = exp lF(0)log(-r-0)'M + exp / 



J i; J it'' 



where F and f are functions of free from infinities between a and />. 



55. The forms obtained in the last three articles suggest the possibility that the 

 addition of Schwarzian factors or the products of Schwarzian factors to one another. 

 or of curve-factors to one another, or of members of the one class to members of the 

 other class, may generally lead to results which are themselves curve-factors, provided 

 all the terms so added have the same angular range. Probably further limitations 

 would have to be introduced into the enunciation of such a theorem to make it valid, 

 but it seems clear that the method of addition is a useful means of obtaining fresh 

 forms of curve-factor. 



The following examples suggest themselves : 



^49 = (w-a)~ + (--c 1 ) m + (w-c 2 ) m + ... + (ir-b)'", . . . . (147) 

 where 1 > TO > 0, and a > c^ > c a > ...>/;; and 



^50= [f(e)(w-6)"de, ....... (us) 



Jfc 

 where J (0) is positive for values of between a and b. 



VOL. CCXV. A. 3 T 



