VIEVFED WITH RELATION TO THE METHOD OP INVARIANTS. 
523 
character ; 2ndly, determining the numerical relations which connect that very im- 
portant form, perhaps of all of its kind, the most important with the forms comprised 
in the fundamental or constituent scale. These questions I propose to consider more 
fully at a future period. For the present I shall content myself with giving a method of 
forming the constituent scale (without, however, seeking the proof of all the forms 
extra to such assumed scale being linear functions of these comprised within it), and 
with determining the numerical relations between the forms in this scale and the 
Bezoutiant for a limited number of values of m. All the forms which we are seeking, 
besides being lineo-linear quadratics, must also be combinantive invariants to / and 
!p, remaining (as forms) unaltered for any linear substitutions impressed either upon 
the variables or upon the functions containing the variables. 
Art. (67.). I must here premise that if there be any two forms of the same degree 
(and that degree odd) in x andy, a combinant maybe formed from them, which will 
be linear in respect to each set of coefficients*. Thus calling the two functions 
2n 
a,x ^^+ ' + ( 2 w -f- 1 ) a, . x"” . 3/ + (2 w -f ] ) * -3/" + • • • + 1 , 
the lineo-linear combinant in question will be 
T=|a..a,„,-(2n+l).a,.a,,+ (2»+l)2B 
which, using our customary notation, will be of the form 
■Y?2n + 1 “ ^0 
(0, 2»+l)-(2«+l)(l, 2«) + ‘^’‘+^’^'* (2, 2»-l)+&c.+ (-)’‘.— 
«-{- 
As a corollary to this proposition (which, as well as the proposition itself, will be 
needed for the purposes of the ensuing determination), taking any function of an even 
degree in x,;y, F{x,^), there will exist a combinant to ^ and by virtue of what 
has been stated above, which will be Mr. Cayley’s well-known quadri variant to F; 
viz. if ihis will be 
fr c a 1 1 2^(2^-!).. .(^ + 1) , 
«o*“2« ii/ita, 2 • • • ~T — ) • C2 n 
The proposition itself is easily proved ; first, the expression T being expressed entirely 
in terms of quantities of the form (r, s) remains unaltered for linear substitutions 
impressed upon the forms /"and <p; it remains then only to show that T satisfies the 
differential equations to T treated as a mere invariant, viz. — 
* I may add here incidentally (although not wanted for our present purposes) that as a combinant in which 
each set of coefficients enters linearly can always be formed to a system of functions 2 in number of as many 
variables and of any odd degree, so reciprocally can a combinant in which each set of coefficients enters linearly 
be always formed to a system of functions each of the degree 2, of which and of the variables contained in 
them, the number is any odd integer. 
