278 Mr Hicks, On the problem of two [Oct. 27, 



_ -^i—, = — ^ l" ^/ ^/ ; that is to say, an arransrement of precisely 

 F Q OP.FQ' '' * r J 



the same nature as the "object." For, first, the image of /i at P is 



a 



-^ at P' and a line sink thence to = — - . Secondly, consider- 

 OP a ^' 



ing any element of the line sink between P and Q, its image 



consists of a line sink element, situated between P' and Q , and 



a line source thence to 0. Hence from Q' to there is a constant 



line source, the sum of all due to the line sink PQ. But the 



whole line sink PQ = — fJi. Therefore the line density between 



and 0' is - ; but that due to the source at P is — - : so that 

 ^ a a 



on the whole the line density between and Q' is zero. 



Next, to find the line density between P' and Q', consider any 



element dx of PQ at a distance x from 0. From this there results 



(writing 1; = ^): 



(\\ _ vdx. - = a at a distance y, where xy = a^\ 



^ ^ X y 



(2) a line source thence towards = — . 



Let p be the line density at a distance y, then 



- ^ / va\ vdx vdya vdya „ 

 cp = d[- —] + -— = —2 -^ = U ; 



y/ a y y 



therefore p = constant ; 



or from Q' to P' there is a constant line density. The magnitude 

 of this can be at once determined from the consideration that 

 there must be no flow through the sphere, that is, that the whole 



amount of the line sink between P' Q must be equal and opposite 



1 



to the source at P' . Hence it must be — -pyy Typ- 1^- 



Since OP . OF = a", this may be expressed -^77^7 • - • Through- 

 out we shall treat these mass-images as single wholes. 



