Jolt — Integrals depending on a Single Quaternion Variable. 13 



For a Yector variable p, (22) reduces to 



Q = llF,{p,^Ap^'p) (37) 



and (31) to 



SO = -JJi?;(p, V)S8pdpd'p, (38) 



so that 



Q/= Q-mi^2(p, V)S8pdpd'p; (39) 



but a direct proof of the relation (38) by Hamilton's method is 

 probably quite as short for anyone not thoroughly familiar with the 

 notation of this paper as the process of deduction from the general 

 result. This last result includes Green's theorem. 



Art. 8. — Finally, so far as quaternions are concerned, we have 

 triple integrals of the type 



Q = \llF{q,l^,^'q,^"q-\) (40) 



in which (compare Arts. 5 and 6) the three independent differentials 

 dg', d'^, d''^" enter combinatorially or in terms of the three-symbol 

 array [d^- d'^' d"^']. The limits of this integral being fixed, exactly 

 as in Art. 5, we may reduce 8 Q to the form 



S Q = m {\F{q, [d^, d'^, cl"^]) - ,\F{q, m, d'^, ,rq-]) 



+ d',i^(^, [d^, 8^, d"^]) - ^\F{q, [d<z, d'g, 8^])} ; (41) 



and because for any quaternions i we have identically 



p {abed) = \hcd~\ 8ap - \^acd'] Shp + [aid^ &cp - [aic^ S dp (42) 



we obtain in terms of D by relations such as (10) the simple 

 equivalent of (41), 



8Q = JliFiq,D).iBqdqd'qd"q). (43) 



The conclusions of Art. 2 and of the last Article apply to this 

 case, the difference of two triple integrals corresponding to two 

 different modes of passage between fixed limits being expressible as a 

 quadruple integral. The condition that the triple integral (40) 



^ Here (ahcd) = Sa{_bccl^ &c. is the single four-symbol array for four given 

 quaternions. 



