Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 6)

ON THE CUBICAL DIVERGENT PARABOLAS. 
103 
399] 
for the equation in X; the quadrinvariant I is =0, and hence the discriminant, 
= — 27»/-, is negative; that is, the roots are two real, two imaginary, as already 
mentioned. 
Considering the simplex forms, first, if c = 0, then for the curve 
y 2 = a? + 2d, 
it appears that R lies at infinity, I within the curve; and for the curve 
y 2 = ofi — 2 d, 
that R lies without the curve, I at infinity. 
It further appears that when d = 0, or for the curve, 
y 2 = a? + 3 ex, 
R, I lie equidistant from the vertex, R without, / within the curve. 
Hence in the curve 
y- = x 3 + Sex + 2d, 
since, when d = 0, the points R, I are equidistant from the vertex, and for c = 0, the 
point R is at infinity, it is easy to infer by continuity that the points R, I lie R 
without, I within the curve, / being nearer to the vertex. 
And similarly in the curve 
y n - = a? + 3 ex — 2d, 
that the points R, 1 lie R without, I within the curve, R being nearer to the vertex. 
Again, in the curve 
y 2 = a? — 3cx + 2d, 
since, in the curve y 1 = a? + Sex + 2d, R is without, I within the curve, and as c 
becomes = 0, R passes off to infinity, it appears that c having changed its sign, or 
for the curve now in question, R having passed through infinity, will be situate within 
the curve; that is, R, I lie each of them within the curve. 
And similarly for the curve 
y- — a? — Sex — 2d, 
it appears that R, I lie each without the curve. 
Hence, finally, for the simplex forms, we have the 7 species of Plucker, viz. 
y 2 — x? — Sex + 2d, c 3 < d 2 , 
simplex ampullate, R, I within the curve; 
y 2 = a?— Sex - 2d, c 3 < d 2 , 
simplex campaniform, R, I without the curve; 
y 2 = sc 3 + 2d, 
simplex neutral, / within the curve, R at infinity;
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.