PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS. 77 



But 



Y.VV.tcIt = ^/tS.tV - rS.drV , 



and 



d(rSrV) = (ItS.tV + rS.^rV . 



These give 



fPdr = - i(rSrV - V.VrrfrV)P = dsY.JJvVP . 



Hence, for a closed curve of any form, we have 



fPdp =ffdsY.TJvVY, 



from which the theorems of §§ 11, 13 may easily be deduced. 



15. The above are but a few of the simpler of an immense host of theorems 

 which any one with some knowledge of quaternions may easily work out for 

 himself, by developing a little farther, or applying to other combinations, the 

 processes just explained. I shall, therefore, give no more of them until I have 

 an opportunity of, at the same time, showing their ready applicability and great 

 value in physical investigations. 



Appendix, added June 3d 1870. 



16. At the instance of Prof. Kelland, to whom this paper was referred, 

 I append a slight sketch of some of the properties of the operator V, of which 

 so much use has been made in the foregoing paragraphs. Most of the results 

 now to be given have been already published by myself, but the mode in which 

 they were formerly deduced has been abandoned for one more purely quaternionic. 



17. It may perhaps be useful to commence with a different form of definition 

 of the operator V, as we shall thus, if we desire it, entirely avoid the use of 

 ordinary Cartesian co-ordinates. For this purpose we write 



S . aV = — d a , 



where a is any unit-vector, the meaning of the right hand operator (neglecting 

 its sign) being the rate of change of the function to ivhich it is applied per unit 

 of length in the direction of the unit-vector a. If a be not a unit-vector we 

 may treat it as a vector-velocity, and then the right hand operator means the 

 rate of change per unit of time due to the change of position. 



Let a, /3, y be any rectangular system of unit-vectors, then by a fundamental 

 quaternion transformation 



V = - aSaV — /3S/3V - ySyV = ad a + fidp + yd y 



which is identical with Hamilton's form given above. (Lectures, § 620.) 



18. This mode of viewing the subject enables us to see at once that the effect 



VOL. XXVI. PART I. X 



