326 The Rev. Thos. P. Kirkman.^;^ 



our unaltered I^+j, determined by our unchanged S, a 

 largest subgroup J^-v+u (^>o)> of m — v-{-i substitutions. 

 These we find in 1^+,, and name them where they stand. So 

 that our index group is now, (^>o), 



i,+i= 1 4-e'i + e'2+ ... + e'^_y+0i + 02+ ... + 0,_„,+„. (14) 



We care not to enquire whether or no, if ^=0, our 

 new 0's, and therefore our new 0's, are identical with our 

 former (6) in J^+i and G^. For even if they are the same, 

 the equivalents 



^i^dd^r^i 0'iGcid02\ &c., 

 that we shall have to handle and mark out, will not be the 



same that 



^i^d^i^j 02Grf67\ &c., 



were in equation (12), because we know that G^and Gd^are 

 different groups in (AJ. 



Our new standard G^^ next prefers his charges against 

 t — m + v of our unmarked equivalents in (A„), from the 

 equation 



for every in I^+i. These are verified by inspection and 

 recorded by ^,^ as before by dy in the proper places. Gdddy 

 Gdddd ' • • n^3.y in turn become standards, and do the like 

 on the yet unmarked groups in (A„* *). By the increase of v 

 we get at last 7n — v = o. There is now no O in I^+i ; no 

 unmarked group in (AJ has more than unity in common 

 with It^i, which is now 



. I,+i=i +01 + 02+. .. + 0f (15) 



We choose any unmarked group G^ for standard, and 

 have to hear / charges brought by it, one for every in 

 (15), from the equation 



(BD = 0i(0r'6n0i) = 0i6E, (16) 



against groups yet unmarked in (AJ. The requisite out- 

 markings are made by D in the proper places. 



