JoLY — Integrals depending on a Single Quaternion Variable. 9 



have the integral Q^ for the second mode connected with the integral 

 Q for the first by the relation 



Q,= Q-\-\hQ= Q+\^^\h,F{q,c\q)-^,F{q,hq)}. (9) 



The limits of the double integral are fixed and prescribed by the 

 modes of passage for the single integrals ; and if the single integrals 

 are single-valued (their modes of passage being given) the value 

 of the double integral is independent of the manner in which the 

 variation has been performed : in other words, the double integral is 

 independent of its mode of passage, provided always that no infijiite 

 terms arise. More generally even if the single integrals are multiple- 

 valued, the double integral is independent of its mode of passage 

 provided that mode is included in a determinate domain. 



Art. 3. — Introducing a quaternion operator D, analogous to V, 

 which operates on q alone, we may write symbolically 



S, = SS^D, d, = Sd^D; (10) 



and therefore we may may replace (9) by 



Q, = (3 + Jji^(^, d^SS^D-S^Sd^D), (11) 



in which we repeat D operates on q alone. It will be noticed that 



d^SS^D - d^Sd^D 



vanishes for hq = ^q and is consequently expressible in terms of 

 the arrays^ 



(d^8^) - 8^Sd£-d^SS(? and [d?8^] = YVd<^ VS<? . (12) 



In fact 



d^SS^D - S^Sd^D = - (d^8^)SD + SVD(d^S^) -YYD[d|?S^], (13) 



and in order that the integral (2) should be independent of the mode 

 ■of passage we must have 



- F{q, ^-a)^!) + F{q,^D {P-a)) - F{q,NYDY a[i) = (14) 

 for all constant vectors a and /8 as we see by replacing the vectors 



1 Trans. Ed. A., vol. xxxii., p. 17. 



