74 PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 



is tantamount to saying that, as the constituents of cr are the potentials of 

 certain distributions of matter, &c, those of t are the corresponding densities 

 each multiplied by 4:n. 



If, therefore, r be given throughout the space enclosed by 2, cr is given by 

 this equation so far only as it depends upon the distribution within 2, and must 

 be completed by an arbitrary vector depending on three potentials of mutually 

 independent distributions exterior to 2. 



But, if cr be given, t is perfectly definite ; and as 



Vo- = v-v , 



the value of V -1 is also completely defined. These remarks must be carefully 

 attended to in using the theorem above : since they involve as particular cases 

 of their ajoplication many curious theorems in Fluid Motion, &c. To these, 

 however, I shall not further allude here, as I propose to make them the subject 

 of a separate communication to the Society. 



11. We now come to relations between the results of integration extended 

 over a non-closed surface and round its boundary. 



Let cr be any vector function of the position of a point. The line-integral 

 whose value we seek as a fundamental theorem is 



f S . a(h, 



where r is the vector of any point in a small closed curve, drawn from a point 

 within it, and in its plane. 



Let cr be the value of o- at the origin of r, then 



Cr = cr - S(tV)ct 



(Proc. R.S.E., 28th April 1862 ; see also Appendix to this paper), so that 



fS.a-dr =/S.((t — S (tV) cr Q )dr . 

 But 



fdr = , 



because the curve is closed; and (Tait on Electro- Dynamics, § 13, Quarterly 

 Math. Journal, Jan. 1860) we have generally 



/S.rVS.cr/Zr = 1S.V(tSo- t - aJY.rdr) . 



Here the integrated part vanishes for a closed circuit, and 



^/Y.rdr = dsJJv , 



where ds is the area of the small closed curve, and U*> is a unit-vector perpen- 

 dicular to its plane. Hence 



/S.o-//r = S.Vo- Vv.ds. 



