172 



ellipse at right angles to the radius vectors through the centre occurs 

 incidentally in Tortolini' s memoir " Sulle relazione," &c., Tortolini, 

 vol. vi. pp. 433 to 466 (1855), see p. 461, where the equation is 

 found to be 



an equation which is obtained by equating to zero the discriminant of 

 a quartic function. Tortolini remarks that this equation was first 

 obtained by him in 1846 in the 'Raccolta Scientifica di Roma,' and 

 he notices that the curve is known under the name of Talbot's curve. 



According to my method, the equation of the curve is obtained by 

 equating to zero the discriminant of a cubic function, and the equa- 

 tion of the surface is obtained by equating to zero the discriminant of 

 a quartic function. 



The paper contains a preparatory discussion of the curve, and the 

 surface is then discussed in a similar manner, viz. by means of the 

 equations 



t,=Y{2-.l(X 2 +Y 2 +Z 2 )j, 



which determine the coordinates x, y, z of a point on the surface in 

 terms of X, Y, Z, the coordinates of a point on the ellipsoid. The 

 surface, which is one of the tenth order, is found to have nodal conies 

 in each of the principal planes, and also a cuspidal curve. The case 

 more particularly considered is that for which 2 > 2b 2 , tf > 2c 2 , and 

 2 -j_c 2 :^ 36 2 , and the memoir contains a figure showing the form of 

 the surface for the case in question. The equation of the surface is 

 obtained by the elimination of X, Y, Z between the above-mentioned 



X 2 Y 2 Z 2 

 equations and the equation 4. ^ + -^ = 1, as already remarked. 



This is reduced to the determination of the discriminant of a quartic 

 function, and the equation of the surface is thus obtained under 

 the form I 3 27J 2 =0, where I and J are given functions of the co- 

 ordinates. 



