232 



PROCEEDINGS OF THE AMERICAN ACADEMY 



^2 

 but of strength — times as great as 3/'s strength, situated at the 

 ar 



inverse point with respect to the circumference of the point where 

 JJ/lies. 



The flow function in this case is easily found, by a method analogous 

 to that used in Section b, to be 



,2 



X ^ f x^ + y'^ — a^ 7^ 



a" 



x^ + y^ - 



a' 



a \[{x- a') + y^Y <^'' \L_t\2., f^ 



(25) 



The combination in the same manner of two pairs of motors of the 

 kind just mentioned yields still another case of flow inside a circular 

 disk due to a certain motor at a point Q, the co-ordinates of which are 

 a and 0, and another motor at the inverse point of Q with reference 

 to the circumference. The flow function in this case is 



a- 



7? + y^ — c? 



[(X - af + /]^ 



A 



a;2 _j. ^2 _ 



{x (x — a)^ + y^ (x — 4 a)} 



+ 



a' 





and this process might be carried on indefinitely. 



12. The image of a simple quadruplet in the circumference of a 

 circle drawn in its own plane is not another simple quadruplet, but 

 a more complex motor. 



On a straight line which passes through the centre of a circum- 

 ference of radius r drawn in a plane, and which is taken for axis of x, 

 let there be two equal and opposite doublets of strength fi with axis 

 coincident with the line, at points dis,tant respectively a and o + Aa 

 from the centre of the circumference. Let there be also on this line 

 the two images of these doublets with respect to the circumference. 



The value at the point (x, y) of the flow function due to these 

 four doublets is 



^y\^x-af+y^ 



{x — a — Acf)'^ -I- y 



+ 



^''^^"'Ai^-v^l+A '^[(^-:7T+^^ 



