On the Existence of Conies loith a Curve of the Third Degree. 463 



as possible, and both the phials be ready at the same instant, and 

 great care taken to avoid the contact of the frogs with the sides 

 of the phials or the liquid. "When all is in readiness, with a pile 

 of two or three elements of Grove, and with an electro-magnetic 

 machine such as is employed for medical purposes, the five frogs 

 suspended on the two iron wires are made to contract. After 

 the lapse of five or six minutes, during which time the passage 

 of the current has been interrupted at intervals in order to keep 

 up the force of the contractions, agitate gently the liquid, with- 

 draw the frogs, close rapidly the phials, and agitate the liquid 

 again. You \d\\ then see that the lime-water contained in the 

 phial in which the frogs were contracted is much whiter and more 

 turbid than the same liquid contained in the other phial in which 

 the frogs were left in repose. It is almost superfluous to add, 

 that I made the complete analysis of the air in contact with the 

 frogs according to the methods generally employed. 



Yours faithfully, 



A. Matteucci. 



LXI. Note on an Intuitive Proof of the Existence of Twenty-seven 

 Conies of closest Contact icith a Curve of the Third Degree. 

 By J. J. Sylvester, Professor of Mathematics at the Royal 

 Military Academy*. 



IN general a conic can only be made to have five coincident 

 points with a curve, and if the curve be of the third degree, 

 the conic will of course cut it in a remaining sixth jDoint ; but at 

 certain points of the cubic all these six points may come together. 

 How many of these are there, and where are they ? This question, 

 which originated with Steiner, who stated the number, and sub- 

 sequently treated by Plucker, who assigned the position of the 

 points, may be resolved by very simple considerations and without 

 calculation. For if we can succeed in putting the characteristic 

 of the curve (I mean what is commonly, but not altogether 

 commodiously, called ''thc-left-hand-side-of-the-equation-to-the- 

 curve- when -the -right- hand-side -of-it-is-made-equal-to-zero") 

 under the form u^ + v{uiv-\-(i>^),'\i is obvious that the conic uw + ai^ 

 will intersect the cubic curve in the six coincident points ?/^ = 0, 

 0)2 = 0. 



If now we take for our cubic the reduced form a^ + r^ + ^ 

 •\-Gmxyz, and make x + y + 2mz = p, px + p^y + 2mz = q, 

 p'^x-\-py-\-2mz = r, it may be written under the form 



{\—%m^):^-\-j)qr> say —fi,z^+2iqr ; 

 or, if wc please, under the form 



-fi{z + kpf +p{qr + fikY + 3/xA;y + Sfikz'^). 

 * Communicated by the Author. 



