TABER. — LINEAR TRANSFORMATION. 189 



generated by the repetition of an infinitesimal substitution satisfying 

 this equation. 



Let <^ be a linear substitution of the first kind of the class we are 

 now considering. That is, let 



li/ Q li^ = i2, 

 and let 



The roots of the characteristic equation of (f> are then the squares of 

 the roots of the characteristic equation of ip. Consequently, if — 1 is 

 a root of the characteristic equation of ^, ^/ — 1 is a root of the char- 

 acteristic equation of i}/ ; that is, 



I xj/ — V^ 8 I = 0. 

 But then 



I ./.-I - V^T S I = 1 12 (,/,-i - V^^l 8) Q-i I = I ,/. - V^S I = ; 



and since 



^-1 _ y'lTi 8 = - V^ ^-^ ("A + V^ 8), 

 we have 



I ^ + V^ 8 I = ; 



that is, — V — 1 is then also a root of the characteristic equation of {j/. 

 It is convenient at this point to introduce a term which has been em- 

 ployed by Sylvester. Thus, following Sylvester, I shall say that the 

 nullity of the linear substitution 4> is m, if all the (m— l)th minor? of 

 the matrix or determinant of $ are zero, (that is, the minors of order 

 n — m + 1, if n is the number of variables,) but not all the mth 

 minors (the minors of order n — m). If now the nullity of ij/ — ^/—IS 

 is tn, then, since 



^j, — ^ZTT s = Q. (,/.-! — V^ 8) n-\ 

 the nullity of 



— V^TiA"' (<A + V^^ 8) = ./.-I — V^^ 8 



is also m ; therefore, since | i//"^ | 4= 0, the nullity of i/r-f- V — 1 8 is m. 

 Whence, by the "corollary of the law of nullity," the nullity of 



^ + 8=(ip- V^^l 8) (^ + V^ 8) 



