14 Proceedings of the Royal Irkh Amdeiny. 



should be independent of the mode of passage is simply (compare (15) 



and (33)) 



F{q, D) = (44) 



'or in terms of V , (compare (17) and (34)) 



Ai.(^,l)_i^(^,V) = 0. (45) 



ot 



Eor the general scalar triple integral having for its element 

 S;?[d$'d'^d"^], the condition (45) is 



-4 Si? - S V Yi; = . (46) 



Of 



Art. 9. — At the commencement of the last Article we stated that 

 the triple integral completed the list for quaternions. A quadruple 

 quaternion integral has a four-spread mode of passage ; in other 

 "vvords the quaternion variable receives every possible value included 

 within the given limits, and the mode of passage is incapable of 

 variation. The methods of the present Paper apply however without 

 formal modification to integrals of a variable 



^ = X ^ XyH-t x^ii + . . . + ar„4 (47) 



where the units ^l , ?' 2 . . . ?'„ obey the laws 



«7 = - 1, i,it + iti, = (48) 



and where multiplication is associative. 



Art. 10. — Analytically the conception of the modes of passage for 

 single, double and triple quaternion integrals presents no difficulty. 

 lV"e have only to conceive the variable quaternion to be a function of 

 one, two or three variable parameters. The limits are defined when 

 a single restriction is imposed on the group of parameters for each 

 limit. Thus the limits for a double integral are defined by two 

 quaternions each of which is a function of a single parameter, and 

 for a triple integral^ the limits are two quaternions, functions of two 

 variable parameters. 



It is worth while inquiring whether we cannot assign useful 

 interpretations for the modes of passage and for the limits when we 

 replace §' by ^^ + p and regard t as the time measured from a fixed 



