494 Mr. A. M c Aulay on Quaternions as a 



furnishing only a hint of a particular solution, and the very 

 easy verification that equation (39) really is a particular 

 solution of equation (37) may follow. 



This form of the particular solution is not the same as 

 Thomson and Tait's ['Natural Philosophy/ § 731, equa- 

 tion (19)], and, for aught that appears above, may be a 

 particular solution different from theirs. It is, however, the 

 same as theirs. To verify this, express SFy . UV in terms of 

 F and w~ 2 SFv » ^u, thus : — 



SFy . U<7=SFy . {ua) = — wF + aSFyw, 



SFy . V™= -SFy . O* 3 o-) = w 3 F-3iT0-SFv^ 

 Eliminating SFyw, 



SFy . Uo-= -fuF— Jw-'SFv . V". 

 Equation (39) thus becomes 



247rN(M + N)e=j'jJ^(2M + 3N)wF-Mw- 2 SFv.V^}^ (40) 



the form given by Thomson and Tait. 



Thomson and Tait regard this particular solution as the 

 solution of the statical problem for an infinite solid. In this 

 case some law of convergence must apply to F to make these 

 integrals convergent. Thomson and Tait (' Natural Philo- 

 sophy,' § 730) say that this law is that Fr (where r is#the 

 distance from some arbitrary origin at a finite distance) con- 

 verges to zero at infinity. This, I think, can be disproved by a 

 particular case. Put, from r = to r = a, F = ; and from r = a 

 to r = oo , J?r=r~ n ct ; where a is a constant vector, and n is a 

 constant positive scalar less than unity. Equation (39) then 

 gives for the displacement at the origin, due to the part of 

 the integral extending throughout a sphere whose centre is 

 the origin and radius R(>a), 



M + 3N R 1 -"-^-" 



3N(M + N) 1-n 



Putting P = co , we get e = x> . The real law of convergence 

 does not seem to be worth seeking, as the practical utility of 

 equations (39), (40) is owing to the fact that either of them 

 is a particular integral for a finite body. 



VI. The Transference of Energy through an Electric Field. 

 What follows is generalized in a paper about to be pub- 

 lished. It is given here, as it will probably, even in the 



