TABER. — LINEAR TRANSFORMATIONS. 33T 



The nullity * of 7\ is then at least one, and the nullity of successive 

 powers of Tk increases until a power of exponent /x = m is attained 

 whose nullity is equal to m. The nullity of the (jj. + l)th and higher 

 powers of T\ is then also m. If we designate respectively by 



nil, 1)12, rn^_-i_,m^ = ?n, 



the nullities of 



rp rpi rpn. — x rp^,. 



•^ A.J -^ A' • • • • -^ A J -^ A> 



then 



mi ^ ^2 — nil = .... = m^j. — m^^-i = 1. 



The numbers /^i, [l^i ^tc, may be termed the numbers belonging to the 

 root A. of the characteristic equation of T. 



If now Th the second power of a real transformation of group G, 

 the numbers belouging to each negative root of the characteristic 

 equation of T are all even. These conditions are probably not only 

 necessary but sufficient in order that a real transformation T of group 

 G may be the second power of a real transformation of this group. 



* The nullity of the transformation defined by the system of equations 



x'r — ari^i + ar2a-2 + . . . + a^, iv^-jv r = (1, 2, . . . N), 



is m if all the(7n — l)th minors (the minor determinants of order N—m-\-V) 

 are zero, but not all the rath minors (the minor determinants of order N—m) of 

 the matrix. 



3'ii) a^isi • • • 



®21> a22> • • • 



VOL. XXXI. (N. S. XXIII.) 22 



