572 Mr. C. E. Weatherburn ; Problems in 



which furnishes an interesting physical meaning for the 

 function TL(tq) as 47r 2 times the surface density of the elec- 

 tricity induced by unit charge at q, all the conductors being 

 connected to earth. 



In order to find the relation which the Green's function of 

 the ordinary electrical theory bears to our function G(pq) we 

 observe that since the potential is zero at a point q of one of 

 the conductors 



<j(*q) + $g(qt)tx(t*)Jt = (10) 



Similarly if the external charge is at ft instead of a, 



In (10) take q as a boundary point t, multiply by fju(tft) 

 and integrate over the boundary. It follows that 



§g(ut)/jL(tj3)dt H-'JJ n{tft) fi{tu)g{td)dt d$ = 0; 



and since the second integral is symmetrical in a and ft so 

 also is the first. Hence 



$g(«t)fi(t0)dt=$g(0t)ii{t*)dt 9 . . . (11) 

 which expresses that the potential at « due to the electri- 

 fication induced over the conductors by a charge — at ft is 



equal to the potential at ft due to that induced by a similar 

 charge at a. This is to a constant multiple the quantity 

 usually called the Green's function in electricity. It differs, 

 however, from our Green's function G +1 (a/3) by the term 



</(a/3), i. e. by the potential at ft due to the charge — at a. 

 For on adding this term we have 



g(ft«) + $g(ftt)fjL(t a )dt, 



which, in virtue of (10) , is zero when ft is a point on the 

 boundary; while the expression becomes infinite at a. — ft 

 like giptft). It is therefore the Green's function* G + i(a/3) 

 as defined for the equation (2). The same follows from (9) : 

 for by it the expression may be written 



g(ftu) + ^gifit)B. +1 {tu)dt, 



which by (56) is identical with the function G +1 (/3a), S} 7 m- 

 metrical in a and ft. 



* u Green's functions for the equation A 2 e< — Pm=0, &c." § I. 



