WILSON AND LF:\VIS. — RELVTIVITY. 457 



As an cx:iiuj)Ie of a similar I'oriuula involving a scalar function /, 

 we may take the familiar theorem of iiydrolynamics that the surface 

 integral of the pressure is equal to the volume integral of the gradient 

 of the pressure /. This is usually written as 



j j fndS = f I I grad/r/S, 



but in our notation becomes 



42. All these formulas lead us to suspect the existence of a single 

 operational equation which is valid when applied to scalar functions 

 and to any vector functions whether with the symbol (•) or (x). 

 This would have the form 



f da,{) = (- 1)P r (rfcr(p,i).0) 0, (64) 



J{v) ^(p+l) 



where da^ is the ^^-vector element of a closed spread bounding a spread 

 of 2> + 1 dimensions. We may extend this equation to four (or more) 

 dimensions, and demonstrate its validity as follows. 



It will perhaps be sufficient to give the proof of the formula in case 

 the (p 4- l)-spread is a rectangular parallelepiped with p + 1 pairs 

 of opposite faces. For let 



dcr(p+\) = ^\2Z...p^\dxidx2dx:i . . . dXp+\. 



Then, by the rules for multiplication, 



J^0'(P+i)-O = (— 1)^ / dx-idx^ . . . (/.rp+i k23...p+it^^i T- 



(p+i) t/(p+i)L "•'^i 



— dxidxz . . . dxp+i ki3...p+i dxo- 1- • • 



0X2 



The partial integrations may now be effected upon the right, and leave 



Jd<r(p+i)'C> = (— 1)^ / dcTip), 

 (p+i) ^(.p) 



if it be remembered that k23 ...p+i, — ki3..p+i, . . . are the positive faces 

 perpendicular to ki, k2, . . . 



It will be evident from this mode of proof that (64) is valid both 



