184 Mr Love, On the Theory of Discontinuous [May 4, 



Remembering that O = log (dz/dw) and using (1) we find 



dz 



B 



du u — c 



n 



'(1 - a n u) + V(l - O V(l - u 2 )' 



u — a m 



-Ai.AJJil-aJ) 



.(5). 



To determine the constants observe first that in passing the 

 point a n the argument of u increases by ir — a n , so that the index 

 — Ai . AJ\/(1 — aj) must be 1 — clJtt and the sum of these 

 indices is unity. Also if u = 1 each factor in the product = 1 

 and 12 = log B is zero so that B = 1. 



The differential equation for z thus becomes 



dz _ 1 

 du u — c 



n 



]i-¥)W(i-Q#-4 



1 - ajir 



u-a n 

 as in Mr Michell's paper, p. 401. 



We have also a series of relations of the form 



-Ai. A„, Ai (a n - c) (a n - c t ) (a n -c 2 )... 



(6), 



1 - — " = . 



7T V(l-0 



V(l-0 (a B -a 1 )(a w -a 9 ) 



.(7). 



There are as many of these as there are a's, that is, as many 

 as the number of the c's + 1 and these would suffice to determine 

 the c's and A in terms of the a's. 



In the particular problem worked out in detail by Mr Michell, 

 viz. that of two rectangular corners a and b for which 



1 >b >c >a> — l, 



the above relations become 



Ai b — c _ Ai a — c _ 1 



~ J(l-b 2 ) F^ = ~V(1-^) o7^b~2' 



giving a relation 



(8), 



which determines c in terms of a and b, viz. 



c = 



vq-aO+vq-fr 2 ) 



• (9). 



