186 Mr Pocklington, On the Configuration of a pair [Feb. 26, 



at G^ to a certain value at H, and then diminishes to — oo 

 at /, the boundary in the plane of w = (^-\- 1-^ is that shown 



in fig. 5. Again, along FE log - is equal to the constant 

 log yj y while 6 increases from to tt ; along GH log - in- 

 creases from log Tj to 00, while d is equal to zero ; and along 



HI log - diminishes from oo to log jj- while 6 is equal to tt. 



The boundary in the plane of fl is therefore that shown in 

 fig. 6. 



The formula transforming the boundary in the plane of w into 

 the real axis in that of t is 



w = \ jdt — - — , 



where we have made F' or G', H' and E' or /' correspond to the 

 points t = 0,t = l,t = oo. 



The formula transforming the boundary in the plane of XI 

 into the real axis in that of t is 



if we make F" or G", H" and E" or /" correspond to the points 

 t = 0, t = l and t = cc. On integration this is 



Noting that ft = log jj. when t = 0, and increases by tti when t 

 increases through 1, we find 2)U. = — 1, and v = log — -jt, ^-nd thus 



Since O = log -y- , this gives 



dz _ dz dw 

 dt dw dt 

 _ _ \ ^ + 1 + 2V^ 

 ~ U t 

 and so 



z = -jj{t-\-^s/t + \ogt-h), 



