40 The Rev. W. Roberts on some Applications of a 



published in Tortolini's Annali, vol. ii.*, under the name of the 

 first negative derived surface — a name suggested by its analogy 

 to the first negative derived curve, so called by me in a paper 

 published many years ago in lAowiMe's Journal de Mathematiques, 

 ser. 1. vol. x. 



If we bear in mind that the parallel to a given surface is con- 

 nected absolutely with it, we see at once that, if its equation be 

 obtained (which in some cases may be done with comparative 

 facility by a judicious selection of the origin), we shall have the 

 equation of the first negative derived from the given surface in 

 reference to any origin whatever, a problem the direct solution of 

 which might be a matter of considerable difficulty ; for we 

 have only to transfer the equation of the parallel surface to any 

 point arbitrarily chosen as origin, and then to make in it the 

 substitutions prescribed by our theorem. Again, it follows ob- 

 viously that if we put \x, \y, \z for x, y ,z, and k + ~ V x 2 -\-y 2 -\-z 9, 

 instead of k in the equation of the parallel, the resulting equa- 

 tion will be that of the first negative derived from the parallel 

 in reference to the origin ; and, as we have just remarked, we 

 can also deduce from the equation of the parallel that of the 

 first negative of the parallel in reference to any origin whatever. 

 It may be interesting to verify the theorem by applying it to 

 the obvious case of concentric spheres. The parallel surface to 

 the sphere 



(*-«)* + (?-£)*+ (s-?) 8 :^ .... (I.) 

 is 



(x-u) 2 +(y-/3) 2 +(z- 7 2 ) = (S + k) 2 . 



Hence the first negative of the sphere (I.) in reference to the 

 origin is 



(*-2a) 2 + (2/-2/3) 2 + (^-2 7 ) 2 =(28-f- \/ w 2 + y 2 + z 2 ) 2 , 

 or 



B \Sx 2 + y 2 + z 2 = ct 2 + /3? + ry 2 -S 2 -ux-f3y-yz, 



a surface of revolution of the second degree, the origin being a 

 focus — a well-known result. 



The discovery, in 1859, of the theorem which is the subject 

 of this note, naturally led me to investigate the equation of the 

 surface parallel to the ellipsoid, an equation which, as I believe, 

 had not at that time been given — as I saw that it would enable 

 me to find the equation of the first negative of the ellipsoid for 

 any origin, a problem which had shortly before received a most 

 elegant solution from Mr. Cayley for the case of the centre as 

 origin. I succeeded in obtaining a solution, which I communi- 

 cated to Mr. Hirst in December 1859. Although inferior in 



* See also an abstract of the memoir in Quart. Journ. of Math. vol. iii. 



