Theory of Canonical Porms and of J^ijperddehmmm^, 



I shall now proceed to generalize this remarkable law^ arid'Tp 

 demonstrate the existence and mode of finding 2n consecutively* 

 degreed independent invariants of any homogeneous function of 

 the degree 4w, and oin-\-\ consecutively-even-degreed indepen- 

 dent invariants of any homogeneous function of the degree 4?2 -f- 2; 

 a result, whether we look to the fact of such invariants existing, 

 or to the simplicity of the formula for obtaining them, equally 

 unexpected and important, and tending to clear up some of 

 the most obscure, and at the same time interesting points in this 

 great theory of algebraical transformations. 



In the first place, let me recall to my readers in the simplest 

 form what is meant by an invariant* of a homogeneous function, 

 say of two variables x and y. If the coefficients of the function 

 f{x, y) be called a^b, c , . ,1, and if when for x we put ax + by, 

 and for y, cx-i-dy, where ad—bc=\, the coefficients of the cor- 

 responding terms become a', V, ,»J; andif I (at, Z>, .../)= I («', 6', ».J), 

 then I is defined to be an invariant of/. 



Let now f{x, y) be a homogeneous function in x, y of the 2tth 

 degree, and write 



where f and 97 are independent of a;, y, and lp^mn = 0. 

 Let ^ x'=lx + my 



j .^^ 'i^z=nx-\-py,j'^,ys « 



then d>^ d_ __£_^ do^ yd dy^ 



^dx'^'^dy'^^dxf'dx '^^d^'d^ 

 io inmmLiiohmbnn . ^ ^/ d di/ >=^do 3<^ ^^i'^ Ji r^ 



'^^'d^'df'^'^dP"^' 



wefind,j^5,^Jt:^i - d^ ^ _h ^ i <? '«'^ ^¥i>d ,K:^«9b 



Again, from the equations hetwem/x/, y'^x^y we B^f^Jj^^^i^f^ 



II ,i' n.i' -,OT-h r^^ px'-rm/ _^-f_^^ ,5»3i^9b d)ii-b f>i« 



^ pl-mn "^^ ^^ *vf9Vri03q8^ 



.eJflijn^v.. tmm: Hyperdeterminant constaiit'derivatiVe; ^*^ "^^ ^^^'^^^ 

 3E3 



