356 ON THE MOTION OF 



be equal to the semi-parameters of the principal parabolic sec- 

 tions. 



4. For a hyperbolic paraboloid, the same transformation, 

 changing the sign of the parameter of the principal negative 

 parabola. The origin of the coordinates of the paraboloids 

 will then be at the summit of the axis. 



5. For an elliptical or circular cone, change the sign of the 

 square of the semi-axis corresponding to the axis of the cone ; 

 then make all the semi-axes infinite, but so that that the ratios 

 of the semi-axis first mentioned to the other two may be equal 

 to the ratios of any altitude of the cone to the semi-axes of 

 the corresponding base. 



6. For an elliptical or circular cylinder, make infinite the 

 semi-axis of the ellipsoid corresponding to the infinite axis of 

 the cylinder. 



7. For a hyperbolic cylinder, make a similar alteration, and 

 change the sign of the square of the semi-axis which corre- 

 sponds to the imaginary axis of the principal hyperbolic section. 



8. For a parabolic cylinder, the same alterations as for 

 either of the paraboloids, making infinite at the same time the 

 third proportional to the two semi-axes corresponding to the 

 normal and the infinite axes of the cylinder. 



The values of the cosines of the normal's axe-angles obtain- 

 ed by means of the differential formulas (19) lead to the fol- 

 lowing equations : (29) 



L = kA x , I* } = kApc , 



L = kA'x' , M, = k,By, 



L = kA"x", N, = kfrz; 



where k and k t are respectively equal to 



v/(A*x* + A'V 2 + A"V' 2 ) ' v/(A,V-+-#,V + C,V) " 



The above equations may be so combined with the equations 

 of the surface, as to furnish other forms for k and k t namely 



