QdT- ^Tviv* -* ^y^i^^stev o» Ajrouhold's Invariants. 



a periodic function in (m) of the second order only fprn^vg 'io od 



oa l+2m— -I+m tii»i-)i) ■uij il has ^{^) 



" l+2»i + 5i— 2m'^^ ^^ :^rrBrrB7ni srfi 



But if we write for a; in the original form px, it becomes b^jooftfi 



^' + 2/' + ^ + ^^''^^^ mvnodJo 00018 



and If for a; we wnte p% it becomes .IdmmmoMmi 



-AM ^4-2^ + ^^ + 6p2m^y^. ! -^^^^^^ 



Hence we can by linear substitutions obtain from a^ + 

 ■i-Qnuvyz the three additional forms 



.(^•W)d> miOl 3lij q . O . q ^/^/ V 



nnoladt 1. .; «^ + 2^' + ^ + 6^(m)a^^, 

 -9'iq oilj M. -"'^;H^i'a?*+2/^ + ^^ + 6S(m)a?y^, 



6w ,woii\i: ,lri-2m' J^ ^ " l'h2pm~l^2pm'' ^«^"i -^ -^ 

 -qwa bfw ^($^()I (va) nt ^l-X'J^^V /,— ^ ' ''-^''^^^^ X^« idbianoo 



In air, there will be twelve values of m forming three remarlcabte^ 

 compound cycles, iJ77 



xisdw lannfiffi ^5l^^ ^i^^^)> lip^), ^ijn), .«— »«) ^Ltiifijjptjgfloo 

 Bfi.«+¥(%-w) /o»2, /o/S(m), /37(m), pg(;„)/ ^' r fuiol 3dno si ^ 

 .floo mw 01 -^ iSJ i,2^^ ^^(,^)^ ^.y(^)^ ^.3(^^ ./ b«a , lo^oalB 



It would be beside my present object to seek to develope fullY,]^ 

 the functional relations in which the several terms of these cycles 

 stand to one another : the interesting relations 



yh{m) = hy{m) =^{m) -, ,-{ 7: 



8/9(»i)=y85(m)=7(w) 



have been already stated by me in another place (Camb. and 

 Dub. Math. Journ., March 1851 *). 'H 



The (S) of the canonical form corresponding to the S of th« ''^ 

 general form is m— m'* ; and the (T) corresponding to the T of the 

 general form is l-20m3-8w«. (See my Calculus of Forms, 

 Camb. and Dub. Math Joum., Feb. 1852.) It is my object to 

 show that any other invariant (I) to the canonical form must be 

 a rational function of S and T. urn 7; isjij ^i dniriv/ 



In the first place, I observe that every invariant to finyftihdi^^' 

 tion of an odd degree i of any odd number q of variables must '= 



• Vide Addendum. 



