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



Any constant may be added to the value of P x . The additional terms thus 

 introduced must vanish. This gives, as in § 9, 



fffV 2 Tch=ffS(UvV)rds. 

 As another verification, suppose r constant, and we have 



ffS.aTJvds = 0, 



which is obviously true. Interesting results are obtained by treating this by 

 the processes of §§ 8, 11. 



25. From one of the theorems above — viz., 



fffS(aV)rds =f/TS.aUvds, 

 we have by the formula of § 17 



ff/Vrd,=ffVv.rds, 



a considerable extension of the fundamental theorem of § 3, which is, in fact, 

 only its scalar part, It might have been obtained, however, as the reader will 

 easily see, by a much more direct process. The vector part 



ff/YVrds =/PfUv.rds , 



as we see by the meaning of VVt in § 21, is of great importance in physical 

 applications, especially in connection with Electricity and with Fluid Motion. 

 When 



T = VP, 



where P is is a scalar, the left hand member vanishes, and the value of the right 

 hand member limited to a non-closed surface is then found as in § 14. 



26. Again, let 



?x = P 2 , 

 which gives 



VP, = - 2p , 



V 2 P X = 6 . 

 We have 



- 2fffS(pV)rds = -fffp^rd, +ffp*${VvV)Tds 



= ~ Sfffrds - 2/frS.pUvds . 



Now if the constituents of t be homogeneous functions of p of the n ih degree, we 

 have for any one of them 



S.pV£ = -«£, 



so that under these circumstances 



(n + 3)fffrd, = -ffrS.pUvds . 



