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



Then, in addition to the condition of homogeneity there given, we have the 

 condition 



V 2 £=0, 



and the general equation of the section referred to gives 



2nfff^U=ffp^.VvV^ds, 



so that, with the help of the final equation of § 26 we have for any closed sur- 

 face whatever 



ff&.Uv{2np£ + n + 3 P 2 Vg)ds = . 



This integral, whose value is obviously the same for all surfaces bounded by 

 a given closed curve, can be reduced to the form 



#(T P ) 4 ^s.u,v(V S^-W 



(Tp)" + 3/ 



where q is any quaternion which satisfies the condition 



Vq = (). 



This is susceptible of various remarkable transformations, both as a double and 

 as a single integral. But this digression might be indefinitely extended, and 

 perhaps has already gone too far. 



30. The essential basis of the whole of this theory is the great invention of 

 Hamilton, by which it is made possible to represent as a vector-operator the 

 square root of Laplace's operator 



d? d*_ d^ 



dx 2 dy % dz 2 ' 



which has not yet been done by any but quaternion symbols, at least in a sym- 

 metrical, easily intelligible, and practically useful form. 



It is rash to make any definite assertions on such matters, especially when 

 a writer of such extraordinary fertility, knowledge, and power as Sir W. R. 

 Hamilton is concerned, but to the best of my knowledge the greater part of 

 the results given above is my own. Hamilton's treatment of V, so far as I am 

 aware of its having been published, will be found in Proc. R.I. A., 1846 and 

 1854, (in the latter of which there is a very curious and interesting proof of 

 Dupin's Theorem,) and in his Lectures on Quaternions, § 620. My own is to 

 be found in Quarterly Math. Journal, October 1860; Proc. R.S.E., 1861-2, 

 1862-3 ; and Elementary Treatise on Quaternions, §§ 317, 319, 364, &c, 418, 

 421-8, Ex. 24 to Chap. IX. and 10 to Chap. XL 



