78 PROFESSOR TAJT ON GREEN'S AND OTHER ALLIED THEOREMS. 



of applying V to any scalar function of the position of a point is to give its vector 

 of most rapid increase. Hence, when it is applied to a potential u, we have 



Vz« = vector-force at p. 



If u be a velocity-potential, we obtain the velocity of the fluid element at p ; 

 and if u be the temperature of a conducting solid we obtain the flux of heat. 

 Finally, whatever series of surfaces is represented by 



u = C, 



the vector Vw is the normal at the point p, and its length is inversely as the 

 normal distance at that point between two consecutive surfaces of the series. 

 Hence it is evident that 



S.dpVu = — du , 

 or, as it may be written, 



— S . dp V = d ; 



the left hand member therefore expresses total differentiation in virtue of any 

 arbitrary, but small, displacement dp. 



19. To interpret the operator V.aV let us apply it to a potential function u. 



Then we easily see that a may be taken under the vector sign, and the 



expression 



V(aV)u = V.aVtt 



denotes the vector-couple due to the force at p about a point whose relative 



vector is a. 



Again, if o- be any vector function of p, we have by ordinary quaternion 



operations 



V(aV).cr = S.aVVa- + aSVa - VSacr . 



The meaning of the third term (in which it is of course understood that V 

 operates on o- alone) is obvious from what precedes. It remains that we 

 explain the other terms. 



20. These involve the very important quantities (not operators such as the 

 expressions we have been hitherto considering), 



S.Vo- and V.Vo- , 



which occur very frequently in the preceding paper. There we looked upon cr 

 as the displacement, or as the velocity, of a point situated at p. Let us now 

 consider the group of points situated near to that at p, as the quantities to be 

 interpreted have reference to the deformation of the group. 



21. Let r be the vector of one of the group relative to that situated at p. 



