619 
S . <j cr- represents the consequent cubical compression of the group 
of points in the vicinity of that considered , and 
V. < cn represents twice the vector axis of rotation of the same 
group of points. 
Similarly 
s -«=-(4 + 4 + .^) = - D '' • • (7)> 
or is equivalent to total differentiation in virtue of our having 
passed from one end to the other of the vector <r\ 
The interpretation of V . cn<| is also easy enough, but it is not 
required for the present investigation. 
II. Suppose we fix our attention upon a group of points which 
originally filled a small sphere about the extremity of g as centre, 
whose equation referred to that point is 
T« = e ........ . (8). 
After displacement g becomes g + er 9 and by (7) g + w becomes 
g -f w + cr-' — (S . w <| )<r‘. Hence the vector of the new surface which 
encloses the group of points (drawn from the extremity of g + is 
W, — <50 — (S . W <J )cj-' (9). 
Hence w is a homogeneous linear and vector function of w l ; or 
& = <pc*) l 
in Sir W. R. Hamilton’s notation, and therefore by (8) 
(10), 
the equation to the new surface , which is evidently a central surface 
of the second order , and therefore , of course , an ellipsoid (Cauchy — 
Exercises , vol. ii.). 
We may solve (9) with great ease by approximation, if we 
remember that TV is very small, and therefore that in the small 
term we may put cq for u — i.e. omit squares of small quantities ; 
thus, 
to = Wj + (S .to i <1 )cn (11). 
Or if we choose we may obtain the exact solution very easily. 
Operating on (9) with S . i, S .j, S . k, we get 
Sito L — Sw^i + <] &c. = &c. 
4 o 
VOL. IV. 
