42 Proceedings of the Royal Irish Academy. 



distribution of doublets, the surface integrals being replaced by sums for 

 isolated doublets, and by volume integrals for polarised matter in bulk. 



It will be noticed that the special moments and products of inertia 

 here introduced are derivable from the ordinary moments and products of 

 inertia for unpolarised matter, 2 m £ 2 an d 2 wi> j£> &e., by replacing each m by 

 the operator ad/dE, + j33/3»j + y3/3? corresponding to a doublet whose com- 

 ponent moments are (a, (3, y). 



15a. With a view to illustrating the utility of the formulae of the 

 preceding Article, let us carry a stage further the approximation for the 

 hydrodynamical case discussed in §§ 13 and 14. The method there employed 

 can be extended without difficulty to terms of a higher order of smallness, 

 but the following is more concise. 



Considering two simultaneous liquid motions, one outside S and specified 

 by <p, the other inside S and specified by $', we take a point P outside S 

 and note that at P for the former motion equation (7) is applicable, while 

 for the latter motion an equation corresponding to equation (3) holds good 



Eemembering, however, that W and W are in opposite senses so that 

 W f W = 0, and that the double-sheets <j> and <£' also have their strengths 

 reckoned positive in opposite senses, we change our conventions so as to 

 make the outward sense of the normal to S the positive sense in both cases ; 

 so we get 



iifp = V'($)- V{W), (21a) 



= - V($')+ V(W'), (21o) 



whence, on adding, 



47T0 = V($ - ^'). (21c) 



Thus the velocity potential is equal to the gravitation potential due to a 

 double-sheet of strength (^ - ^)/4ir. 



At great distance, therefore, <p is given approximately by formula (17), 

 provided (£ - ^)/4jt be substituted for r in formulae (18) and (19). 



The dynamical significance of (p, q, r) in this connexion has already been 

 discussed. It is clear that a, b, c, f, g, h, are likewise simply expressible in 

 terms of the impulsive pressure, but not in terms of the impulsive forcive 

 from without. 



An alternative expression, not involving </>', may, of course, be got by 

 using formula (21a) and taking the approximations to V (</>/4tt) atid V ( W'\i.tr), 

 the latter by the known formula for an unpolarised distribution, the former 

 by formula (17). 



16. Kelvin's Inversion Theory applied to doublets. — It is a well-known 

 theorem, due to Liouville, 1 that the most general conformal space-transfor- 

 1 Journal de Mathe'matiques, t. xv, 1850, p. 103. 



