422 Mr. R. F. Gwyther. On the Differential [May 4, 



These equations will be abbreviated to 



1 /=Q 1 / (A). 



0,/=Q 2 / (B). 



3 /= QJ (C). 



§ 2. — Interpretation of the Conditions (A), (B), awZ (C). 



Write r4> q (7r—p) r (g—x)? as a type of the terms in the covariant. 

 Then (1) 0,f = Q x f becomes 



r <fr q = Q lr+1 2 _!/g, 



so that, in any set of terms homogeneous in (vr—p) and the 

 coefficients may successively be derived from that of the term in 

 (77-— p) r+ 2 by means of the operator Q x . 

 And (2) 2 / = Q 2 / becomes 



r4>q = —ihr<f>q+lld — r — q, 



so that the coefficient of such term homogeneous of degree r + q in 

 (it— p) and (£— may be derived from the coefficients of the set of 

 terms homogeneous of degree r + g + l in (»-— p) and (f — a?). 

 Also (3) 3 / = 3 / becomes 



r0 2 = — Q 3 r+i0 ? /d — r— 5. 



We thus have a second mode of derivation of character similar to 

 the last, showing that the operators are not all independent. 

 The order of derivation is shown by the chart 



d&o *~ d<fii — ^ rf02 — ^ 



\ / \ / \ 



^-i0o ^ d-\<pi d-\(th ^ 



\ / \ / 



J-200 ^ rf_ 2 0l ^ , 



where — >■ denotes deduction by the operator 



in which it is easy to trace the relations 



QiA — QxQa = O 

 Q,Q 3 -Q 3 Q 2 = 0, 

 Q 3 Q A — QiQa = 0. 



Q 



