JoLY — Integrals depending on a Single Quaternion Variable. 7 



of these theorems due to Tait and proved by him and other -wTitors in 

 various "ways. Indeed, Hamilton regards the subject as one of great 

 difficulty and dismisses it rather abruptly ; but his method is of wide 

 scope and merits further developments. 



I propose therefore to sketch some of the consequences of 

 Hamilton's method in relation to quaternion integrals depending on 

 a single quaternion variable, and from certain general results I shall 

 deduce as particular cases the extensions of the theorems of Stokes 

 and Green. It is not my object to furnish short proofs of these 

 theorems ; they can be readily supplied from the results of this Paper 

 by substituting from the commencement vectors instead of quaternions 

 and by integrating round closed curves or over closed sui'faces. 



In. the concluding articles it is shown that the quaternion integrals 

 are capable of physical applications, and the more concrete character 

 of these articles may assist in forming a clearer conception of the 

 natiire of the general integrals considered in the earlier portion. 



As the integrals discussed in this Paper depend essentially on the 

 combinatorial functions which I have called quaternion arrmjs (Trans. 

 R.I. A., vol. xxxii., p. 17), it maybe useful to recapitulate the formulae 

 which we shall require. (Compare " Elements of Quaternions," 

 Art. 365 (6)). 11 a, h, c and d are any quaternions, the arrays are 



{al) = Yb8a- Ya S5 ; [a5] = Y.YaYb; {ale) = Sa [ho'] ; 



[ahc'] = {ahc) - \lc] ^a - [_ca'] SJ - [«5] Sc 



and 



{abed) = 8a[bcd^. 



Transposition of contiguous symbols changes the sign of an array, 

 and an array vanishes if its constituents are linearly connected. 

 Also for any fifth quaternion e, 



a{icde) + b{cdea) + c{deab) + d{eabc) 4 e{abcd) = 

 and 



e{abed) = \bcd'\ Sae - \acd'\ S5e + \abd'\ 8ce - [«5c] 8de. 



Art. 1. — If F{q, r) is any function of two quaternions, distributive 

 with respect to the second, so that 



F{q, r-^s) = F{q, r) + F{q, s) , (1) 



