70 DEFINITE INTEGRALS. [INT. III. 



If M be a factor to be determined, we may put 



(5) MX = A, MY=B, MZ=C, 

 where A, B, C are the above determinants. 



But the determinants A, B, C, if differentiated by x, y, z, 

 respectively and added, are found to satisfy identically the solenoidal 

 condition 



(6) :, + *i-+f^ 



dx dy dz 

 so that we have the equation for M, 



d(MY) 

 (7) dx dy 



Consequently for any continuous vector function R it is possible 

 to find a scalar multiplier M that shall make the vector whose 

 components are MX, MY, MZ, solenoidal. If the vector R is 

 itself solenoidal, the equation for M is satisfied by any constant, 

 say 1, so that in this case we have 



y _ d\ d/JL d\ dfi 



~~ dy dz dz dy ' 



^ = 8\a/^_8\a/A 



dz dx dx dz ' 



_ 9X 3//, d\ d/j, 

 dx dy dy dx ' 



But if PX, P M denote the vector parameters of the functions 

 X, IJL we see by the definition of the vector product, 



R 



If we consider two infinitely near surfaces of the first family 

 for which X has the values X and X + d\ respectively, the normal 

 distance between which is dn^, we have by 16 and 20 



Considering two infinitely near surfaces of the other family 

 and fj, -t- dp, we have in like manner for their normal distance 



, d/j, 



