'ii 



TRANSACTIONS OF SECTION A. 



647 



number of terms in anyone of tlie groups G,, G.^, Sec, the c sum of a cycle of 

 Hn»»Vu'P m"' roots <if unity IK a rutionnl fHnctiv)t of \\ and v., (§ 17). Finally, Mb 

 law of K ran her, that the equation F(.i') =0 is an Aljeliun when a certain e.vpre8,^ion 

 p the root of an Ahelian equation of the (ni-l )"' degree, ii taken as known, is 

 deduced, the exact nature of pi beiny detertnined (§ 18). 



5. The Tactiurariarit of a Conical Section and a Cubic Carve. 

 By Profossor F. Lixdkmann, Ph.l). 



T)i sign by uS' = i/' = c^' = . , . a binary quantity of tlio yixtli order, and by 



A, B, C, A„„„ llio invariants of it as they are defined by the following formulas (cf. 

 (Jlebsch's ' Theorie der binaeren algebraischen Formen') : 



i ^ ' ^ i f i 

 / ' = («(■) '«/, m"' = (//)■■/-, v/ - = (miy-i^', 



A = (ahy; 15 = (//',', ( ' = (ii'/(i'i'y{i"ir, 



A„,m =("""')'"'• 



According to Mr. Brioscui, the discrimina it of a " is given by the expression 



i''-;3-A-' + 23-;!-0-"A-B -f 2'-5'-A-C + 2-;3'5--AB-'+ 2--a-o'-BC + 3--o'A„„„. 



From this one may derive the tact invariant of two planer curves, one of the second, 

 tlio other of tlie third order, by a method vvliich I liave established in the 

 'Bulletin de la Societo Mathematique de France' (t. v. et vi.). .Suppose the 

 e juation of the first one given by 



and the equation of the cubic by 



() = « " = & "^r =>=.,.. 



X X J^ 



The tactinvariant of those two ternary quuntics is evidently not changed, if we 

 take the quantic 



n ■' = (B-^ = A« •' - l{aj)'p")-a_ . j) '; 



instead of a ^ i where A = (pp'j>")' ) ', nnd so we have to do if we follow the method 

 referred to. 



The simultaneous invariants of ^' '•' and a ", wanted for our expression, may be 

 introduced by the following system of forniukc : 



/ ;- = {nl3p)\l3, II ■■ = {aliyfafy^, 



/ , = (I I/y/)-II_^,, e^u^ = («/3«)-'« ,^.,. 



A = (/>_//)■, r = (ee7)'>S7>.?7, 



8 = {aPy)ia[id.){aym^y8), 

 T = (a/3y)(a^ll)(ayll)(/'i)/ll). 

 Suppose now that a point ,r of the conic j> - = is represented by the para- 

 meter li : ^.,, so that 



and put 



B = {hk')(h'k"){k"k); 



then one has oD' = A (cf. loc. cit.), and tlie farther application of our method leads 

 to the results 



