Theorem relating to Parallel Surfaces. 45 



I adopt the form of the equation of the parallel given by M. 

 Catalan (Nouvelles Annates, vol. iii. p. 555) ; and by writing in 



.11- 7 -, x y k „ 



it a , ^ tor a, o, and ^ + y9 _ A * tf + yi-tf' a-a + y*-** t0r ^ 



y, and A, I find for the equation of the parallel to 



\a 2 b 2 -{a 2 + b 2 )(x 2 + y 2 -k 2 )\ 2 \a 2 x 2 + b 2 y 2 -{a 2 + b 2 )k 2 ^(x 2 + y 2 -k 2 ) 2 \ 

 + M 2 \x 2 + y 2 -k 2 \\a 2 b 2 -{a 2 + b 2 )(x 2 + y 2 -k 2 )\ 3 

 -27a*b A k\x 2 + y 2 - A; 2 ) 2 + 18a 2 6 2 * 2 \x 2 + y 2 - k 2 \ \a 2 b 2 

 - {a 2 + b 2 ) [x 2 + y 2 -k*)}{ a*a? + b 2 y 2 - (« 2 + 6 2 ) * 2 ~ (* 2 + 2/ 2 - * 2 ) 2 } 

 + 4^ 2 {«V + ^V-(« 2 + ^)^ 2 -(^ 2 + ?/ 2 -A: 2 ) 2 } 3 = 0. 

 Now, put x 2 -\-y 2 =k 2 in this equation, and it becomes 



\ a 4 b 4 [a 2 y 2 + h*x*f - 4a 2 b 2 {a 2 y 2 + &V) 3 = 0. 



Dividing by a 2 b 2 (a 2 y 2 + b 2 x 2 ) 2 , and writing Jo:, |y for <r ; ?/, we 

 get for the first negative of a 2 x 2 + b 2 y 2 = (x 2 + y 2 ) 2 , 



a 2 ^ b 2 ~ ' 



as indeed is evident. We thus obtain an interesting verification 

 of our different theorems. 



Again, taking from Salmon's ' Higher Plane Curves/ p. 279, 

 the equation in elliptic coordinates of the involute of an ellipse, 

 namely, 



we have for the first negative of the involute the equation 



Mr. Salmon has derived the equation of the surface parallel to 

 an ellipsoid from the condition that a sphere should touch the 

 ellipsoid. The condition, in general, that a sphere should touch 

 a given surface is expressed by a relation between the coordinates 

 of the centre of the sphere, its radius, and the constants in the 

 equation of the given surface. If in this relation we suppose 

 the radius of the sphere to be constant, and regard the coordi- 

 nates of the centre as variable, we have the equation of the 

 parallel surface. It is evident also that the above relation will 

 give the locus of the centre of a sphere tangent to the given sur- 

 face, and the radius of which is a given function of the coordi- 

 nates of the centre. If we suppose that the equation of the 

 parallel surface has been obtained, such a locus will be had by 



