278 UNIVERSITY OF COLORADO STUDIES 



Designating the slope of the direction of B by m,^ the equation 

 of the cubic in Fig. 1 becomes 



^ 4:[af-\-y^) — 1 4(.'B''+y) — 1 



By a collineation which transforms the quadruple of Fig. 1 into 

 another orthogonal quadruple, i. e., a quadruple in which BjB^Bg are 

 the foot-points of the altitudes, this equation assumes the form 



{axJr^y) {x'+f) +aii?+2hxy+cf+2dx+2ey+f=0, 



which is the equation of the general circular cubic. Again, by col- 

 lineations any cubic may be transformed into a form whose equation 

 has the form 



,/^a{x—e,){x-e^){x—e,), 



with '^i^gj^fj. According to the values of «j, e^^ 63, the cubic may 

 belong to one of five classes as first established by Newton. O 



I. The cubic serpentine with oval (parabola campaniformis 

 cum ovali). 



The 6's are all different from each other and all real. 



II. The cubic serpentine (parabola pura). 



f^, is real, and e^ and e^ are conjugate imaginary. 



III. The cubic serpentine with isolated point (parabola 

 punctata). 



e^ = e^ different from e^ and all real. 



IV. The nodal cubic (parabola nodata). 

 e^ different from €^=e^ and all real. 



Y . The cuspidal cubic (parabola cuspidata). 



fi=e^2 = 6'3 and all real. 



This somewhat lengthy introduction will be sufiicient for the 

 understanding of the constructions of the five types as they follow 

 from the Steinerian transformation. 



I. The Cubic Serpentine with Oval, 



This cubic is obtained when all four points of the fundamental 

 quadruple are either real, or imaginary. As the case of four real 



(■) Enumeratio linearum tertii ordinis (Londini, 1706). 



