( fi: 



to 1)0 (leieniiiiied by moans of (T)— (Al), a problem lluii may bo 

 reduced to equations of the fonn 



1 ö^p 



^"'^^W = -'' ^1) 



ill Avhioli <c is a known, and if' an iniknowii rnnclion ai' .i\ i/, :,t. 



Lot tt bo any olosod sni-faco and // llio normal lo ii, drawn onl- 

 wards. 



Then, if the equation (I) holds in the whole spaoe >S', enelosed 

 by o, we shall have for the value of if" in a point J* of this space, 

 at the time t, 



Here the tirst intei>ral extends over the space S and the second 

 over the boundary surface a-, ;• is the distance to /■*, and the square 

 brackets serve to indicate the values of tlio enclosed (juanlilies for 



the time t . 



c 



Let us now conceive the surface (7 to recede on all sides to infinite 

 distance and let the circumstances be such that the surface-integral 

 in (2) has the limit 0. Then, ultimately: 



1 n 



</(j. 



(2) 



V' 



4ji 



Jt- 



] <JS, 



(3) 



where the iiitogration must be extended over infinite space. 



§ 3. Equations of the form (1) may be deduced from the formulae 

 (I) — (VT) in many ditforont ways; they may e.g. be established for 

 each of the comj)onents of ^ and '\ ^) The sobiti(^n is ho^^■(M•or olv 

 tainod in a simpler form '■'), if one inti-oducos four auxiliai'y <|uanlilies, 

 a scalar [)Otential <f and the three components a,., a,^, ci, of a vector- 

 polential a. These quantilies satisfy the o(pialioiis 



1 d'<p _ 



Af/) 



A a,,. 



1 d^Vr 



--Q i\r. 





— Q l\,/, etc., 



so that, with liic restrictions that are re(pnred if (3) is to be true, 

 we nuiA' write 



1 ri 



1) liOiiKNTz, Ija Ihéovie óleclr()ini\giiérK(iic do Maxwf.li, o\ ?on application aux 

 corps mouvants, Arch, néerl. T, i25, p. i7(l 189^. 



~) See Lkvi Civita, Nnovd Cimento, (i), vol. G, p. '.):3 , 1S97 ; Wiechkrt^ 

 Arch, néerl, (2), T. 5, p. 549, 190Ü. 



