TRANSACTIONS OF SECTION A. G47 



number of terms in any one of the groups G,, G 2 , &c, the e ram of a cycle of 

 primitive m ,h roots of unity in a rational function ofv l and r 2 (§ 17). Finally, the 

 laio ofKroneeker, that the equation F(.r) =0 is an Abelian when a certain expression 

 p , the root of an Abelian equation of the (m-1 ) th degree, is taken as knoxvn, is 

 deduced, the exact nature of Pl being determined (§ IS). 



5. The Tactinvariant of a Conical Section and a Cubic Curve. 

 By Professor F. Lindemann, Ph.D. 



Design by « e 6 = i/ = c /= • •• a Dmai 7 quantity of the sixth order, and by 

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

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



i * = (aby a ?b: = i' 4 = »'; 4 = . . . . 

 i* = («o 4 «/; "*/ = C0' J y > »/ = W* f *> 



A = (abY, B = (»') 4 , U = (ii'y(i'i"Wiy, 



According to Mr. Brioschi, the discriminant of a e is given by the expression 

 2 7 -3-A 5 + 2 3 -3-5 3 -A 3 B + 2 4 -5 4 -A 2 C + 2'3'5 S -AB S + 2 2 -3-5 5 \BC + 3 2 -5 5 A w ,„. 



From this one may derive the tactinvariant of two plane curves, one of the second, 

 the other of the third order, by a method which I have established in the 

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

 equation of the first one given by 



and the equation of the cubic by 



= a 3 = 6 3 = c 3 =.... 



XXX 



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

 take the quantic 



«>£ x 3 =a^ 3 -§(«///ov.?v 



instead of a » (where A = (pp'p")~) ; and so we have to do if we follow the method 



referred to. 



The simultaneous invariants of p 2 and a 3 , wanted for our expression, may be 



x X 



introduced by the following system of formulae : 



ij = (afrfajij H , 3 = («/3y) 2 « fy x , 



l x = <Upp')*H x , ejuj = (aW J* J 



x = {ipp'Y, r = (Ge'0 2 pS 2 ^7, 



S = (a£y)(a/3S)(a 7 S)OyS), 

 T = (a0yXa/3H)(ayH)(0yH). 

 Suppose now that a point x of the conic pf = is represented by the para- 

 meter |, : £.„ so that 



7 O T/O 7 / ' -^ 



■f'l = %£ I *2 = k f » ' r 3 = * { t 



and put 



D = {kk')(k'k")(k"k); 



then one has 3D 4 = A (cf. loc. eit,), and the further application of our method leads 

 to the results 



