JoLY — Integrals depending on a Single Quaternion Variable. H 



respect to the two independent differentials d^' and di.'q and satisfies 

 the general condition ^ 



F{q,r,s) + F{q,s,r) = 0. (23) 



The limits being fixed, a variation of the two- spread mode of passage 

 gives 



8Q = li8F(q,dq,cVq); (24) 



and writing as before (4) 



8F{q, dq, d'q) = 8,F{q, dq, d'q) + F(q, 8dq, d'q) + F(q, dq, Sd'q) 

 dF{q, d'q, Sq) = d,F{q, d'q, Sq) + F{q, dd'q, 8q) + F{q, d'q, d8q) (25) 

 d'F(q, 8q, dq) = d',F{q, 8q, dq) + F{q, d'8q, dq) + F{q, 8q, d'dq) 



we find on addition by (6) and (23), 



8F(q, dq, d'q) + dF{q, d'q, 8q) + d'F{q, 8q, dq) (26) 



= 8,F{q, dq, dJq) + d^F{q, d'q, 8q) + d',F{q, 8q, dq). 



The limits being fixed, integration gives in place of (24) the relation 



8Q = ll\8,F{q, dq, d'q) + d,F{q, d'q, 8q) + d',F{q, 8q, dq)}, (27) 



Art. 6. — By Article 3, as a consequence of the relation (23), the 

 function F(q, r, s) must involve r and s combinatorially, that is in 

 terms of the arrays (rs) and [rs^. "We may therefore write 



F{q, r, s) = F, {q, {rs)) + F, {q, [ri\), (28) 



the functions being distributive with respect to (rs) and [rs] re- 

 spectively. Or for the sake of brevity if "we use the notation 



F(q,r,s) = F{q, {rs]) 



instead of the expanded relation (28), we may by (10) replace (27) by 



8Q = \lF{q, {dqd'q]^8q'D + {d'q8q]^dq'Yi + {S^d^jSd'^D). (29) 



As in Article 3 the element under the signs of integration must be 



^ It is apparently impossible to assign any meaning to an expression of the 

 t3rpe of (22) in -wMch. tMs condition is not satisfied. 



