TRANSACTIONS OF SECTION A. 647 



9. On the Symbolism appropriate to the Study of Orthogonal aiid Boolian 

 Invariant Systems which appertain to Binary and other Quantics. 

 By Major P. A. MacMahon, F.B.S. 



It has been customary to study the theory of invariants by considering the 

 invariants of the general linear substitution as of primary importance, and to pay 

 comparatively little attention to the invariant forms connected with the orthogonal 

 and BooUan substitutions. These latter substitutions are particular, and give rise 

 to a number of forms which include those which arise from the general substitu- 

 tion ; so that from one point of view the ordinary invariant theory is a particular 

 case of the theory of orthogonal or of Booliau invariants. This is the view taken 

 in this paper ; the orthogonal and Boolian systems are studied by means of six 

 invariant symbolic factors, and at any time the theory of Clebsch and Gordau 

 and Aronhold can be derived by restricting attention to the two symbolic factors 

 employed by them. 



10. .4 Quintic Curve ca/nnot have more than fifteen real Points oj Inflexion. 



By A. B. Basset, F.R.S. 



Zeuthen has shown ' that not more than one third of the total number of pointa 

 of inflexion that a quartic curve can have can be real, I propose to show that a 

 similar proposition holds good in the case of a quintic curve. 



A quintic curve cannot have more than six double points, nor more than forty - 

 five points of inflexion. 



Let A B C be the triangle of reference ; and let A be a triple point composed 

 of three real crunodes ; and let B be a real crunode. Then the equation of the 

 quintic is 



a-Mj + ay^s + yX = (1) 



where «„, v^, Wg are binary cubics in li and y. 



Since each tangent at the triple point is composed of two real nodal tangents, 

 the three tangents at this point count as six real stationary tangents. If, however, 

 each tangent at the triple point has a contact of the third order with its respective 

 branch (which is the highest contact that a tangent at a triple point on a quintic 

 can have), each tangent counts three times as a real stationary tangent ; and there- 

 fore the foregoing singularity reduces the number of real points of inflexion by nine. 

 In this case, v.^ = ku^, and (1) becomes 



(a- + kay)u^ + y'-iv^ = . . . . . (2) 



If each of the tangents at the crunode B has a contact of the fourth order with 

 its respective branch (which is the highest contact which a tangent at a node on a 

 quintic can have), each tangent counts three times as a stationary tangent, and 

 therefore the foregoing singularity reduces the number of real points of inflexion 

 by six. 



We shall now show that these two singularities can coexist oo a quintic . 



Let 



^ (X, lM,v,^y_ ^, yf 



, = (I, VI, n, py^l3, yY 



then (2) may be arranged in the form 



/33{X(a- + kay) + ly'} + /32y{/i(a« + kay) + my^ 

 + ^y' { v{a- + kay) + wy- } + ■nny^ {a' + kay) + py' = . . (3) 



If the nodal tangents at B have a contact of the fourth order with their 

 respective branches 



/i = pX, m = pl 

 V — crX, n = al 



Math. Aniialen, 1875. 



"■3" 



