Lrathbm — On Doublet Distributions in Potential Theory. 45 



If is situated at a source in the original system so that, near 0, f 

 approximates to the form - mJiirR, then in the inverse system, for great 

 values of R', <p' approximates to the value <p'^ = (- m v j4:nR)(k!R') = - mjink, 

 a constant. Conversely if, for great values of R, <p approximates to a constant 

 value x , there is a source mj = - 4irk<p T at in the inverse system. 



If is situated at a doublet /_*„ in the original motion, so that <f> approxi- 

 mates near to the form - (fu/i^R' 2 /, the most important part of <f>' for 

 great values of R' is - (/i ( ,/4Tr)k~ 3 R'f, so that there is ' uniform flow at 

 infinity ' of velocity v = - (/M,/±Tr)k~ 3 . Conversely uniform flow v at infinity 

 in the original motion gives a doublet fi ' = - inkfv at the origin in the inverse 

 motion. 



20. The possible occurrence of a constant in the limiting analytical form 

 of <j> at infinity is seen to play a perhaps unexpectedly important part in 

 determining the nature of the inverse system. This is not really surprising 

 when one gets accustomed to the fact that the inverse of a field of constant 

 potential (corresponding to liquid at rest) is a field of potential corresponding 

 to flow from a source. In the original motion we can add to <f> any constant 

 we please, making the constant <£„ zero or whatever else we like, without 

 changing the motion. In the inverse motion this gives us at the point a 

 source of arbitrary strength, which may be adjusted to satisfy some special 

 requirement. 



21. The Continuity Condition. — Though the general theory of the inversion 

 method leaves -no room for doubt that the inverse motion is a possible motion, 

 namely that it satisfies a continuity condition of the type of formula (5), it is 

 nevertheless worth while to inquire how the equality in this form is obtained 

 directly by the formulae of correspondence. Let us therefore take each term 

 of the continuity condition of the inverse motion, namely 



jV'f/.S" = [J/'] + 2m' + 2,/ +■ [;«'„], (30) 



and express it in terms of the data of the original distribution. It is to be 

 remai-ked that the term M is present only if the centre of inversion is in 

 the relevant region of the original motion, and the term ?n ' is present only if 

 is in the relevant region of the inverse motion. 



Now WclS 1 = ( JFr 3 /A- 3 - $r* cos x/& 3 ) (A'/r 4 ) *S', 



M' = 4:trlc(j>o, %m' = %m!;/r, 



2"' = - Xifik/r 2 ) cos x, mo = - ^k<p„ ; 

 so equation (30), on division through by k, is the same as 



\WrhlS -\$ cos x r- 2 dS = [4*r<p ] + tmr- 1 - %nr' 2 cos x - [>rfj. (31) 

 If is in the irrelevant region of the original motion the first term in 



