( 69 ) 



IV. — On Green's and other Allied Theorems. By Professor Tait. 



(Received April 29th, read May 16th, 1870.) 



I was originally attracted to the study of Quaternions by Sir W. R. 

 Hamilton's ingeniously devised and most valuable operator 



^j . d . d . d 



dx •' dy dz ' 



to which he called special attention (Lectures on Quaternions, § 620) on account 

 of its promise of usefulness in physical applications. But I soon found that 

 in order that its full power may be applied, in general investigations, it is 

 necessary that we should have processes of definite integration, of the kinds 

 required in physics, applicable to quaternion symbols and not merely to scalar 

 variables. I often consulted Hamilton about this want, and he promised to 

 endeavour to supply it at some future time. I fancy that shortly before his 

 death he must have in some way supplied it, though he certainly did not print, 

 nor does he appear even to have written, anything on the subject. In one of 

 the last letters I received from him, he said that he intended to conclude the 

 final chapter of his Elements, which is devoted to physical applications, by 

 some sections on the use of the operator mentioned above. That chapter 

 remains unfinished, and as Hamilton rarely wrote down the steps of even a 

 complex train of mathematical reasoning until he had mentally completed it, it 

 is to be feared that this portion of his investigations is entirely lost. So far as 

 the analytical aspect of Quaternions is concerned, this loss is very serious 

 indeed, for there can be little doubt that Hamilton's solution would have been 

 of immense value from the purely mathematical point of view. 



I have recently succeeded to a certain extent, by a simple, though not very 

 direct, process, in supplying the want — so far at least as to enable me to use 

 quaternions in inquiries connected with potentials — and have thus arrived at 

 very simple proofs of Green's celebrated theorem and various allied results, 

 some of which appear to be new and valuable. The quaternion calculus can, 

 in consequence, be applied without loss of its enormous special advantages to 

 various general theories, such as Attractions, Spherical Harmonics, Fluid 

 Motion, &c, &c. Curiously enough, I find that I had almost arrived at one of 

 the general theorems given in the present paper so long ago as 1860 (Quaternion 

 Investigation of the Potential of a Closed Circuit, Quarterly Math. Journal), but 



VOL. XXVI. PART I. T 



