VECTOK ANALYSIS. 49 



100. On the interpretation of the preceding formuice. Infinite 

 values of the quantity which occurs in a volume-integral as the 

 coefficient of the element of volume will not necessarily make the 

 value of the integral infinite, when they are confined to certain 

 surfaces, lines, or points. Yet these surfaces, lines, or points may 

 contribute a certain finite amount to the value of the volume-integral, 

 which must be separately calculated, and in the case of surfaces or 

 lines is naturally expressed as a surface- or line-integral. Such cases 

 are easily treated by substituting for the surface, line, or point, a 

 very thin shell, or filament, or a solid very small in all dimensions, 

 within which the function may be supposed to have a very large 

 value. 



The only cases which we shall here consider in detail are those of 

 surfaces at which the functions of position (u or w) are discontinuous, 

 and the values of Vu, Vx, V.o> thus become infinite. Let the 

 function u have the value u^ on the side of the surface which we 

 regard as the negative, and the value u 2 on the positive side. Let 

 Au = u 2 u l . If we substitute for the surface a shell of very small 

 thickness a, within which the value of u varies uniformly as we pass 



Vtt 

 through the shell, we shall have Vu = v within the shell, v denoting 



a unit normal on the positive side of the surface. The elements of 

 volume which compose the shell may be expressed by a[dcr] , where 

 [dcr] is the magnitude of an element of the surface, do- being the 

 vector element. Hence, 



Vu dv v Au [dcr] = Au d<r. 



Hence, when there are surfaces at which the values of u are 

 discontinuous, the full value of Pot Vu should always be understood 

 as including the surface-integral 



[p -/> 



relating to such surfaces. (Au f and dv are accented in the formula 

 to indicate that they relate to the point p f .) 



In the case of a vector function which is discontinuous at a surface, 

 the expressions V.wdv and Vxwdv, relating to the element of the 

 shell which we substitute for the surface of discontinuity, are easily 

 transformed by the principle that these expressions are the direct 

 and skew surface-integrals of w for the element of the shell. (See 

 Nos. 55, 56.) The part of the surface-integrals relating to the edge 

 of the element may evidently be neglected, and we shall have 



V. CD dv = o> 2 . dcr ojj . dcr = Ao> . c&r, 

 V x o> dv = do- x fc> 2 d<r x Wj = d<r x Ao>. 



G. II. D 



