664 PROCEEDINGS OF THE AMERICAN ACADEMY. 



eoci — coc-i = e-2.Cz = 0, ezC\ = ^3^2 = C3C3 = 0: 

 if 



A = f/oCo + 03^3 9^ 0, B = Ci, 



we have 



'Aei=^, CiB = = 1, 2, 3);' 



and we may now put n = rii = 3, and 



ei = en, Cn = €23, ^3 = €13. 



On the other hand, let m = 2 and let (ci, €2) contain a number A 9^ 

 such that 



A ei — A 6'2 = 0. 



In this case, we may, without loss of generality, put .1 = ei, when we 

 have . . 



er = 0, Cic-2 = 0. 

 If now 



ei = fdn^% ^lo^'A {i = 1, 2), 



we then have, since eC~ = 0, 



0„(l), ^„(1A = ^ /O, k 



•' co-l 



where k 9^ and the determinant of the matrix ei is not zero; and, 

 therefore, since eic^. = 0, 



^e,,(^), 0,2(2A = u; (a, ^\ W-\ 



where, without loss of generality, we may put a = 1, |9 = 0, giving 



eoei = Ci, e-f = eo. 

 This system, however, contains no number B 9^ for which 



eiB = e^B = 0. 



Second, the number system (fi, co, ... Cm) may contain either a 

 number A 9^ such that Aci = for / = 1, 2, . . . m, or a number 

 B 9^ such that CjS = for i = 1, 2, . . . m, but not both. In this 

 case, we may put n — m and assign to the 0's and 17's either the values 

 given by equations (90) or equations (88) respectively. 



Third, the system (^i, e^, ... Cm) may contain neither a number 

 A 9^ such that Aei = for i = 1, 2, . . . m nor a number B 9^ 

 such that CiB = for i = 1,2, ... m, for Avhich a sufficient, but not 



