70 PROFESSOR TAIT ON GREEN'S AND OTHEE ALLIED THEOREMS. 



though I then gave a special case I did not see that a very slight modification 

 of my work would have enabled me to generalise it. I was then seeking to 

 derive from my formulae the well-known physical result, and not thinking of 

 extending the calculus itself. 



Even the little advance that I have made in the present paper has enabled 

 me to see, with a thoroughness of comprehension which I had despaired of 

 attaining (at least by Cartesian processes), the mutual relationship of the many 

 singular properties of the great class of analytical and physical magnitudes 

 which satisfy what is usually known as Laplace's equation. This is, of course, 

 solely due to the simplicity and exjDressiveness of quaternions in general. 



1. In what follows we have constantly to deal with integrals extended 

 over a closed surface, compared with others taken through the space enclosed by 

 such a surface ; or with integrals over a limited surface, compared with others 

 taken round its bounding curve. The notation employed is as follows. If Q 

 per unit of length, of surface, or of volume, at the point x y z, Q being any 

 quaternion, be the quantity to be summed, these sums will be denoted by 



f/Qds and ffffyh, 



when comparing integrals over a closed surface with others through the 

 enclosed space ; and by 



f/Qds and fQTJp, 



when comparing integrals over an unclosed surface with others round its 

 boundary. No ambiguity is likely to arise from the double use of 



JfQds , 



for its meaning in any case will be obvious from the integral with which it is 

 compared. 



2. I have already shown (Proc, R.S.E., April 28th 1862,) that, if o- be the 

 vector displacement of a point originally situated at 



p = ix + jy + kz , 

 then 



S.V 



<x 



expresses the increase of density of aggregation of the points of the system 

 caused by the displacement. (See Appendix to this paper.) 



3. Suppose, now, space to be uniformly filled with points, and a closed 

 surface 2 to be drawn, through which the points can freely move when 

 displaced. 



Then it is clear that the increase of number of points within the space 2, 

 caused by a displacement, may be obtained by either of two processes — by 

 taking account of the increase of density at all points within 2, or by estimating 



