PROFESSOE TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 71 



the excess of those which pass inwards through the surface over those which 

 pass outwards. These are the principles usually employed (for a mere element 

 of volume) in forming the so-called " Equation of Continuity." 



Let v be the normal to 2 at the point p, drawn outwards, then we have at 

 once (by equating the two different expressions of the same quantity above 

 explained) the equation 



ff/S.Vack =ff S.aTJv els, 



which is our fundamental equation so long as we deal with triple integrals. 



4. As a first and very simple example of its use, suppose o- to represent 

 the vector force exerted upon a unit particle at p (of ordinary matter, electricity, 

 or magnetism) by any distribution of attracting matter, electricity, or magnetism 

 partly outside, partly inside 2. Then, if P be the potential at p, 



o- = VP, 



and if r be the density of the attracting matter, &c, at p, 



Vo- = V 2 P = 4nr 



by Poisson's extension of Laplace's equation. 



Substituting in the fundamental equation, we have 



krrfffrds = 4ttM = f/S . VPUv ds , 



where M denotes the whole quantity of matter, &c, inside 2. This is a well- 

 known theorem. 



5. Let P and ~P 1 be any scalar functions of p, we can of course find the 

 distribution of matter, &c, requisite to make either of them the potential at p ; 

 for, if the necessary densities be r and ?\ respectively, we have as before 



V 2 P = 47JT , V 2 P X = 477TJ . 



Now 



V.PP 1 = PVP 1 + PxVP, 



and 



V 2 .PPj = PV 2 P X + P X V 2 P + 2S .VPVP X . 

 But, by the fundamental theorem, 



fffV\V?sk =ffS.(V.¥? l )JJvds =^S.(PVP X + ¥y¥)\Jvds. 

 Substituting the above value of V 2 . PP X , this becomes 



f/S.(pV¥ x + P^LWs =fff(PV 2 P 1 + P^P)^ + 2/# r S.VPVP 1 $ ? . 

 But, obviously, we have also by the fundamental theorem 



#S.(PVP X - PiVPJUvtfe =^(PV 2 P 1 - P^P)^, 



