35 



when we put 



D'Vj = 





, 3/3=:J«tan.«: (K*) 



w being the angle which the plane passing through the given ray and 

 parallel to the near ray makes with the plane oixz; and Vi i\ being 

 the extreme values of r. 



The equation (J*) expresses in a simple manner the law of the 

 virtual focus. It shews that the extreme positions of that focus cor- 

 respond to the same pair oi natural coordinate planes, passing through 

 the given ray, which we considered in the preceding number, and 

 which we may therefore call the planes of extretne virtual foci, as 

 well as the planes of extreme projection. Indeed, when the given ray 

 is one of the principal rays of the system, assigned in the fourteenth 

 number, then all the virtual foci, as well as all the other foci hitherto 

 considered, coincide in the principal focus : and the planes of extreme 

 virtual foci become, in this case, indeterminate. However, we shall 

 shew that their place is then supplied by another remarkable pair of 

 planes, which pass through the principal ray, and complete the sys- 

 tem of natural coordinates : but for this purpose it is necessary to 

 enter briefly on the theory of aberration from a principal focus, which 

 we shall do in the following number. 



Aberrations from a Principal Focus. 



19. If we conceive a plane cutting a given ray perpendicularly at 

 a given point, this plane will be nearly perpendicular to the near 

 rays, and will cut those rays in points near to the given point : the 

 distances of these near points from the given point, are the lateral 

 aberrations of the near rays, and the cutting plane may be called. 



