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



Keturning to the quaternion form, as far less complex, we see that 



Vo- = scalar = 477T, suppose, 

 implies that 



cr = VP, 



where P is a scalar such that 



V 2 P = 4tt>- ; 



that is, P is the potential of a distribution of matter, magnetism, or statical 

 electricity, of volume-density r. 



Hence, for a non-closed path, under conservative forces 



-fS.<rd P = -fS.VPdp 

 = ~/&(dpV)? 



= fd dp P =y>/P (see Appendix) 

 - P — P 



depending solely on the values of P at the extremities of the path. 



13. A Vector theorem, which is of great use, and which corresponds to the 

 Scalar theorem of § 11, may easily be obtained. Thus, with the notation already 

 employed, 



/V.adr =/V(o- - S{rV)a )dr, 



= -/S(rV)V.ov/T. 

 Now 



V(V.VV. rdr)a = - S(rV)V.oy/T - S(tfrV)Vro- , 

 and 



d(S (tV) Vo- O r) = S (rV) V. oy/r + S {d T V)Va r . 



Subtracting, and omitting the term which is the same at both limits, we have 



/V. a-dr = V. ( V."LW)a-o ds . 



Extended as above to any closed curve, this takes at once the form 



/V. o-dp =ffdsY. (V.U^V)o- . 



Of course, in many cases of the attempted representation of a quaternion 

 surface-integral by another taken round its bounding curve, we are met by 

 ambiguities as in the case of the space-integral (§ 9) : but their origin, both 

 analytically and physically, is in general obvious. 



14. If P be any scalar function of p, we have (by the process of § 11, above) 



/Vdr =/(P - S(rV)P )r/r 

 = -/S.rVP .f/T. 



