372 Mr. J. J. Sylvester on Aronhold's Invariants. 



2=s6i-f5 ^nd (7=61-1-3 should be combined, aM th6 two c^I 

 eluding paragraphs in page 302 will then read as Allows'*—' 



If <7=62 + 5 or =6f-f-3,/t=i + l j and the factor ()n3--^m)^'*+i 

 is of the degree 6t + 9, t''^^itt'eatih'casegl♦^8e* than; ($)i Which 

 is absurd. , .nr«i:i -rfMV/ '•, f! ,7 :^v\>^\ <\MV:. ^a^i^^n') Jt 



^.fijs i)Jrjo(Irrijlg i[iiv7 



Addendum. r: '?«() lirnfKqmon nC 



OnJhejmtwreofihe three Cycles of four terrhs each whim ()dki^tH 



tkiJwetve values of the parameter to the canonichl'forih'^f^a 



cuilfic function of three variables. "^^ 



^^ "jj^^j ^[jijfttions given in the text show that each term in any 

 AAe cycle is a periodic function of the second order of each other 

 terg^jm'.the same cycle. Moreover, it may be shown that each 

 terWL in any one cycle is a periodic function of the third order ot 

 every tfa'ni in either of the other two cycles ; a sort of relation 

 between the cycles taken per se, and with one another precisely 

 ihe inverse of what obtains (as already shown) for the two cycles 

 Of ;thi'ee terms, each containing the six values of the parameter 

 to the biquadratic function of two variables. For as regards 

 that case, it was shown in the first part of this paper that the 

 terms in the same cycle are periodic functions of the third order 

 of one another, and of the second order of each of those not in 

 the same cycle with themselves* 

 If we make ^^^'^^ 



^^on.n»A 5~^=B f^^^-^C ^+3^:^^ .bm 

 .^^^pA=A' pB=B' pC = C,'' ;^ _ ^^=^.bhofria 



The following table will exhibit all the ternary periods that 

 can be formed between the terms of the several cycles : — 



(1) A B' D" (4) B A' C" (7) C A' D" (10) D A' B" 

 ^^(2) A C B" (5) B a J)" (8) C B' A'' (11) D B' Q" 

 ']^($) A D' C" (6) B D' A" (9) C D' B" (12) D C \% 

 .^{(jf or instance, as an example of the meaning of the table, take 

 line (8), viz. C, B', A". This indicates that A" is formed from 

 B' and C from A" in the same way as B' from C, and of course 

 A" from C in the same way as C from B' and B' from A'^ &c. By 

 means of this table it will easily be seen that a term in each of 

 two cycles being given, the term in the third which forms with 

 the given two a ternary period may immediately be assigned. 



The remarks which I have to add on the nature of the equa- 

 tions for finding the parameter (m), as well for (.r, ?/)'* as for 

 (jSj y, zY, will be given hereafter. 



O*-' ^^^ ^ continuedj ^^^j j^^.j^^jj, ^^^ ^^^^ ^^ 



liiol ) haudi II« htm ^:iol(l .iM ^(d 



