Quaternions in Physics. 95 



space. This gives at once 



where JJv is unit normal drawn outwards from the bounding 

 surface. If we put for a the expression uyv, where u and v 

 are any two scalar functions of position, this becomes Green's 

 Theorem. 



If the space considered be imagined as bounded by two 

 indefinitely close parallel surfaces, and by the normals at each 

 point of a closed curve drawn on one of them, this is easily 

 reduced to the form of the line and surface integral 



§8",TJvv<rd8=$8<rdp, 



The simplest forms of these equations are respectively 



and 



§Y(Uvy)uds=$dpu, 



where u is any scalar function of position. But it is clear 

 from the mode in which it enters that u may be any quater- 

 nion. And it is easy to build on these an indefinite series of 

 more complex relations. Thus, for instance, if a and t be 

 any two vector-functions of p, we have 



which has many important transformations. You will find it 

 laborious, but alike impressive and instructive, to write this 

 simple formula in Cartesian coordinates. It consists of four 

 separate equations, containing among them 189 terms in all ! 



In the three relations just given we have the means of 

 applying quaternions to various important branches of mathe- 

 matical physics, where Nabla is indispensable. But I must 

 confine myself to one example, so I will take very briefly the 

 equations of fluid motion. 



Let e be the density, and a the vector-velocity, at the point 

 p in a fluid. Consider the rate at which the density of a little 

 portion of the fluid at p increases as it moves along. We 

 have at once, for the equation of continuity, 



which we may write, if we please, as 

 g=Sv(«r). 



