16 Frederick C. Ferry. 



power of X or [JL is made possible by increasing its degree 

 in V , it will not be possible in general to effect any decrease 

 in f. Since k' represents the degree in X , ix^ of the terms 

 of to .'z> lowest in X" and ijl , as to . cp is arranged for substitu- 

 tion, k' can be decreased only by increasing by 2x (x a posi- 

 tive integer) tlie degree of v in o) . cp ; but such change in uj 

 increases f by 2 and m' by one ; hence, so long as m' remains 

 a minimum, neither f nor k' can be altered. Therefore, so 

 long as m' remains a minimum will f, k', and g' + gp + h' -|- 

 h^ -|- 1' -f Iq be invariants peculiar to the curve (p , q) whose 

 equation is <p = 0. 



To show how the substitution may actually be performed 

 in to.cp; — 



Let any term of œ . cp be 



C', x^^yß V- 



-a' -|- |3' -f- t' = P + n' , e-oen. 



Y' < n^ 4" C[ ) and since p — q + n' — n^ > n^ 4- q 



(cf. p. 12), then must 



T'<K°^' + 13' + t') or 

 t' < a' + ß'. 



"We may now have any one of these three cases, viz., — 



1) l<Y'<a' and ß' even, whence 



C'. X^'fx^'vT' = C'.xTU^^°^^-Asaß^ 



2) 1 < y' < a' — 1 and ß' odd, whence 



C'. l''ii^\'l' = C'. xT' - 1. y . zK°^' - Y' - 1). s^(ß^ - 1). 

 Here t'4=°'' since ß' is odd and consequently a' -[- t' is odd 



