44 Proceedings of the Royal Irish Academy. 



From this it is but a step to the idea of corresponding surface dis- 

 tributions, namely an original distribution consisting of a double sheet of 

 strength t = ^ <iS and an inverse distribution consisting of a double sheet of 

 strength -' = fijdST together with a surface density zs = — ft cos \ r'dS : 

 that is to say r = -7 



■d = - rJ: cos \ V 2 . 



Here - and -' are regarded as of the same sign when in the corresponding 

 senses of normals to the inverse surfaces, that is when they make supple- 

 mentary angles with r. 



18. Hydrodynamical Illustration of Inversion. — Let us now consider the 

 hydrodynamical system described in § 5 for which the formulae (3) and 

 hold good. "We have a surface S separating a relevant from an irrelevant 

 region, and in the relevant region liquid sources and doublets: and cor- 

 responding to these we have a gravitation system comprising a surface 

 density IT, a double-sheet *, particles m, and doublets y.. 



If we invert all this we get a surface S separating a relevant from an 

 irrelevant region, and in the relevant region we have particles typified by 

 hi' = mr'/k and ••' = - u I r"- /. J ) cos x, doublets u = u" : : -~ ; a surface 

 density TT = : cos \. and a double-sheet of strength 



9 = c . These satisfy the relation analogous to (3) 



r\ 6 ■ = r ■ ■ W, - - _ (28) 



since each side is kS 1 - 1 times the corresponding side of equation (3) at 

 the corresponding point. Farther, for the relevant region, there is a function 

 f' (namely -.!;■ rii„e5 9 at the corresponding point; such that 



4tt 9 ' =- V : - ■'. TV, -m. -, .29) 



It follows that 9' specifies a liquid motion corresponding to a normal 

 velocity W of the boundary S. 



19. The taking of the centre of inversion in either of the relevant regions, 

 with the consequent extension of the inverse relevant region to infini ty, may 

 give rise to peculiarities in <p or <p requiring special examination. 



If is in the relevant region of the original motion, and is not a point of 

 singularity of <p, then at a small distance R from the approximate form 

 of 9 is ^o +/E where / is a spherical surface harmonic of order unity. Con- 

 sequently in the inverse region the approximate form of o' for great values of 

 E : K'" 1 (<p - and so the inverse system has a " source at 



infinity " of strength M = 4=7:^. Conversely if the original motion has a 

 " source at infinity " of strength M the value of q,' at in the inverse system 

 = .. 4* 



