346 Prof. Cayley on the Theory of the Evolute. 



different terms, and in particular in order to exhibit the destruc- 

 tion of the term 6 2 in the coefficient of Y. But the equation is, 

 it will be observed, a quartic equation in #, with coefficients 

 which are linear functions of the current coordinates (X, Y, Z). 



The equation shows at once that the evolute is of the class 

 4; in fact treating the coordinates (X, Y, Z) as given quan- 

 tities, we have for the determination of an equation of the 

 order 4, that is, the number of normals through a given point 

 (X, Y, Z), or, what is the same thing, the class of the evolute, 

 is =4. 



The equation of the evolute is obtained by equating to zero 

 the discriminant of the foregoing quartic function of 6 ; the 

 order of the evolute is thus = 6. There are no inflexions, and 

 the diminution of the order from 4.3, = 12, to 6 is caused by 

 three double tangents. 



I consider the particular case where the conic touches the 

 absolute. There is no loss of generality in assuming that the 

 contact takes place at the point (y = Q, z=Q), the common tan- 

 gent being therefore £=0; the conditions for this are a = 0, 

 h = 0, and we have thence C = 0, F = 0. Substituting these 

 values, the equation contains the factor 6 ; and throwing this 

 out, it is 



X(-H0 3 + (B + 2G)<9 2 ) 



+ Y( A0 3 - 2H0 2 ) 



+ Z( - A <9 2 +3H0-(B + 2G))=O, 



or, what is the same thing, 



03(-H X + A Y ) 



+ 2 ((B + 2G)X--2HY-A Z) 



+ 0( 3H Z) 



+ ( -(B + 2G)Z) = 0, 



where it will be observed that the constant term and the coeffi- 

 cient of 6 have the same variable factor Z, where Z = is the 

 equation of the common tangent of the conic and the absolute. 

 The evolute is in this case of the class 3. It at once appears 

 that the line Z = is a stationary tangent of the evolute, the 

 point of contact or inflexion on the evolute being given by the 

 equations Z = 0> (B + 2G)X — 2HY = 0. The equation of the 

 evolute is found by equating to zero the discriminant of the 

 cubic function ; the equation so obtained has the factor Z, and 

 throwing this out the order is =3, The evolute is thus a curve 



