114 



Mr. A. L. Selby on two 



Images, due to Mr. Hicks, are required. His proofs will be 

 found in the ' Philosophical Transactions 'for 1880 ; but in 

 §§ 2-4 of this paper I give other proofs, in which the subject 

 is considered from a slightly different point of view. 



The main interest of the problem of pulsating spheres is 

 derived from its experimental illustration by Prof. Bjerknes. 



§ 1. Preliminary Definitions. — A source of liquid of strength 

 m is a point at which a volume of liquid 47rm is supplied per 

 unit time. A sink (or negative source) of equal strength is 

 a point at which liquid disappears at the same rate. A com- 

 bination of a source and sink of equal numerical strength rn } 

 at an infinitesimal distance d apart, is a doublet. As this is 

 analogous to an infinitely small magnet, I shall call rnd the 

 moment of the doublet. The intensity of a doublet is its 

 moment divided by its length ; it is analogous to the intensity 

 of magnetization of a bar of unit section. The axis of a 

 doublet is the line from the sink to the source. 



The Hues of flow due to any distribution of sources and 

 sinks are the fines of force due to the corresponding magnetic 

 poles. 



§ 2. Let P, P f be a source m and sink — m on the straight 

 line OL: 



TPL = 6 ; TP'L = & ; TP = r ; PP' = d. 



Then the flow through the circle (of which TT' is the pro- 

 jection) formed by revolving T round OL is 



2777ft (cos & — cos 6). 



Fisr. 1. 



Let P approach P so that PP' diminishes indefinitely, 

 md retaining a constant magnitude //,. Then 



2irm (cos 0' — cos 6) = 2tt/* sin 2 6 . jr. 



If the source and sink are interchanged, the sign of the flow 

 is reversed ; we may consider yu as negative in this case. 



