330 The Rev. Thos. P. Kirkman on 



definition and identification. Thus the work goes rapidly 

 on. 



There is an inelegance in the repeated out-markings 

 which cannot be avoided in the symmetrical handling of the 

 subject in perfectly general terms ; but in practice this 

 inelegance completely disappears. In all the definite cases 

 that I have studied, of ;^<9, the Index-group can be so 

 written as a product of two or more groups, that, by using 

 not all the 0's, but only the substitutions of a factor group, 

 we can avoid all repetition of outmarkings. The inelegance 

 affects not the validity of the preceding symmetrical general 

 demonstration. 



On these transformations of the Index-group, it would 

 be useless to say more until we have a definite table (AJ 

 before us. I have worked through many such tables, and 

 hope to present the results to this Society. 



14. I may now exhibit my new theorem. Let G^^ be our 

 standard group, and 0„ be any substitution of the index- 

 group determined by our 2, which ^„ is not a substitution 



of G^. Let 



Ge = 0aGA-' («) 



be an equivalent of G^ which contains, not 0^5 but an equiva- 

 lent 0ajm+i^«^ of the subgroup 



Jm+i = I + 61 + 62 + . . . e„, (Wl> 0), 

 which is common to G^ and our index-group I^+i. 



Let GJ • • and G^ * * stand for the two lines in our table 

 (A J, which exhibit G^ and G^ with their Q - i derivates, and 

 let Q>t ' ' and (B? * ' stand for those lines under 2, viz., for the 

 functions (3^ and (5^ followed by their Q - i values. Then 

 is 



G^* • = Gg + lAgGe + 2AeGe + ' /-lAgGg +/AeGe 



+y+iAeGe+" + y_iAeGe = N,; (d) 

 which by (a) is 



G+- • = 0aGA-' + iKOaG^e-^ + ' ' +/_iA,0«Gd0^^ 



+fKe,GaO-^ +f+iKeaGaO-' + • • • = Ne ; W 



