PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 73 



8. If, in the fundamental theorem, we suppose 



o- = Vt , 

 which imposes the condition that 



S.Vo- = 0, 



i.e., that the a displacement is effected without condensation, it becomes 



ff&.VrVvdp =fff&.V\ch = 0. 



Suppose any closed curve to be traced on the surface 2, dividing it into two 

 parts. This equation shows that the surface-integral is the same for both parts, 

 the difference of sign being due to the fact that the normal is drawn in opposite 

 directions on the two parts. Hence we see that, with the above limitation of 

 the value of a, the double integral is the same for all surfaces bounded by a 

 given closed curve. It must therefore be expressible by a single integral taken 

 round the curve. The value of this integral will presently be determined. 



9. The theorem of § (4) may be written 



fffV 2 ¥ds =ffS.TJvV¥ds = J r S(UvV)P<fe. 

 From this we conclude at once that if 



o- = iP + /P, + kV 2 , 

 (which may, of course, represent any vector whatever) we have 



ff/V^ck =ff&{UvV)<Tds t 

 or, if 



W = T, 



fffrd,=ff^(VvV-')rds. 



This gives us the means of representing, by a surface-integral, a vector-integral 

 taken through a definite space. We have already seen how to do the same for 

 a scalar-integral — so that we can now express in this way, subject, however, to 

 an ambiguity presently to be mentioned, the general integral 



where q is any quaternion whatever. It is evident that it is only in certain 

 classes of cases that we can expect a perfectly definite expression of such a 

 volume-integral in terms of a surface-integral. 



10. In the above formula for a vector-integral there may present itself an 

 ambiguity introduced by the inverse operation 



v- 1 



to which we must devote a few words. The assumption 



VV = T 



VOL. XXVI. PART I. U 



