12 Proceedings of the Royal Irish Academy. 



a function'of the three-symbol arrays^ [S^, d^, d'^'] and (8^, d^, d'j). 

 In fact the relations 



(Ic) S ap + {ca) ^Ip ■\- {ah) 8cp = \_p, [«5c]] 



\ ) 

 \lc] Sap -^ [cfl] 8hp + [c3] Sep = - {p, [aicj) 



may be proved -without any difficulty, so that we have 



SQ = IJ^i {S, [D, [S^d^d'^]]} -IIi^2 {?, (D, [S^d^d'^])}. (31) 



Art. 7. — Thus, given any tTvo modes of passage for the variable 

 of the double integral (22) between the fixed limits, if it is possible 

 to pass from the first to the second by continuous variation without 

 introducing infinite terms, the difference of the values of the double 

 integral is expressible as a triple integral whose limits are prescribed 

 by the two modes of passage, and, except in cases of multiple values 

 of the double integrals, the value of the triple integral is independent 

 of its three-spread mode of passage. 



If the double integral (22) is independent of the mode of passage, 

 the element of the integral (31) must vanish, or replacing [S^d^d'^*] 

 by an arbitrary quaternion a we must have 



F,{q,lI>a^}-F,{q,{J)a)} = 0, (32) 



or separately for the scalar and vector part of a, 



F,{q,YD) = 0, F,{q,Y\Da)-SJ)'Fo{q,a) = (33) 



a being an arbitrary vector. Or in terms of V and ^:— by (16) this is 



C 



F,{q,V) = 0, F,{q,YVa) + ~F,{q,a) = 0. (34) 



The general scalar double integral is of the form 



J/S(ri(d^d'-^)+JJSo-2[d^d'?], (35) 



and for this the conditions reduce to 



SVo-,. = 0, ^- = VDo-i. (36) 



of 



1 These arrays are defined by the relations 



{abc) = SYaYbYc; [aie] = {ahc) + [c5] Sa + [«c] Sb + [iff] Sc, 



in which a, b and c are any quaternions, and as {abc) = S \_abc] any three-symbol 

 array can be expressed in terms of [aJe]. 



