VECTOR ANALYSIS. 45 



92. If u or CD is a scalar or vector function of position in space, 

 we may write Pot u, New u, Pot CD, Lap o> for the volume-integrals of 

 pot u' y etc., taken as functions of p ; i.e., we may set 



Potu=fffpot u'dv' =fff. ?' , dv', 



' [p ~p]o 

 / 



New u = fffuew u'dv' = /77V^ ^ u'dv', 



' [p -pti 



- t 



IP -/Jo 



where the p is to be regarded as constant in the integration. This 

 extends over all space, or wherever the u' or o>' have any values 

 other than zero. These integrals may themselves be called (integral) 

 potentials, Newtonians, and Laplacians. 



QQ d Pot u _ .p du d Pot ft) ^ , dw 



t/o. T - IT ot -5 , T -- = Jrot -7 



ax ax ax ax 



This will be evident with respect both to scalar and to vector functions, 

 if we suppose that when we differentiate the potential with respect 

 to x (thus varying the position of the point for which the potential 

 is taken) each element of volume dv' in the implied integral remains 

 fixed, not in absolute position, but in position relative to the point 

 for which the potential is taken. This supposition is evidently allow- 

 able whenever the integration indicated by the symbol Pot tends to a 

 definite limit when the limits of integration are indefinitely extended. 

 Since we may substitute y and z for x in the preceding formula, 

 and since a constant factor of any kind may be introduced under the 

 sign of integration, we have 



V. Pot co = Pot V. 



i.e., the symbols V, V., Vx, V.V may be applied indifferently before 

 or after the sign Pot. 



Yet a certain restriction is to be observed. When the operation of 

 taking the (integral) potential does not give a definite finite value, 

 the first members of these equations are to be regarded as entirely 

 indeterminate, but the second members may have perfectly definite 

 values. This would be the case, for example, if u or w> had a constant 

 value throughout all space. It might seem harmless to set an inde- 

 finite expression equal to a definite, but it would be dangerous, since 



