relating to " Canonic Roots " 455 



or, neglecting as premised all considerations of algebraical sign, 



/ 3^ \ 2(n-l) 



= fan- 1) 2 \Tr~ ) - ft P^»- a x ra _ 2 ) , 



z e. 



R(X W , X n _!j R(X n _ 1 , X w _ 2 ) R(X W _ 2 , X w _< 



\2(n-l) \2fft-2) \2(n-3) 



= &c, = R(Xp Xq) = 1; 

 or if any one of my readers finds a difficulty in admitting that 

 H(ax— b, l)=a, he can stop short at — ^- 2 -| — — > which may 

 easily be verified to be equal to unity. Hence 



R(x w ,x n _ 1 )=xf- 2 .; Q.E.D.; 



Thus we see that if X„, X re _j have one root in common, \ n must 

 vanish ; but then, by the cited theorem of Jacobi, it follows that 

 X n completely contains X w _ x ; from this it was easy to infer the 

 necessity of the function* of which X w is the canonizant, having 

 infinity for one of its " canonic roots " — or, in other words, of its 

 being reducible to the form 



And so it became natural to establish a priori the existence of 

 this condition, and thus to obtain the proof virtually reproduced 

 by Mr. Cay ley in the article referred to. 



In what precedes, X n-1 was a first principal minor of X w ; and 

 it occurred to me to institute an inquiry into the form of the 

 resultant of two functions related to each other as X„_! is to X„, 

 with the sole but important difference that the constants in X^ 

 are not to be contained in a concatenated order from one line to 

 another, but to be taken perfectly independent as in the example 





1 



' X 



X 1 



X 5 





x 3 = 



a 

 a' 



b 



c 



d 

 dt 







a" 



b" 



c" 



dn 





satisfying the identity U=(Aa?-fB2/)V+2/ 2 W; we have then 

 R(U, V)=R(V, y*W)=R(V, W)X (R(V, y)) 2 , 



where evidently R(V, y) is the coefficient of a;"- 1 in V ; let y become unity, 



then on calling U, V, R(V, y) X n , Xn-i, Xn-i respectively, and giving to 



W its corresponding value, we have the theorem as it is used in the text. 



* For in general if X n be the canonizant to F, Xn- 1 will be the canoni- 



, , d 2 F 



zant to -— . 



ax 2 



