106 



a formula that evidently coincides with the one that we found before, for the height of the os- 

 culating focus. 



IX. On thin and undevelopable pencils. 



[41.] Having examined^ some of the most important properties of the developable pen- 

 cils of a reflected system, we propose in this section to make some remarks upon pencils not 

 developable ; and we shall begin by considering thin 'pencils, that is pencils composed of rays 

 that are very near to a given ray ; because in all the most useful applications of optical theory, 

 it is not an entire reflected or refracted system that is employed, but only a small parcel of the 

 rays belonging to that system. 



To simplify our calculations, let us take the given ray for the axis of (2), and let us choose the 

 coordinate planes as in the preceding paragraph ; the cosines of the angles which a near ray 

 makes with the axes of (r) and [y], will be, nearly, 



it i^ 



X, y being coordinates of the point in which it meets the mirror; and the equations of this near 

 ray will be, nearly, 



x = ^ + «2i', TZ — y-t i8z', 

 that is 



x' = «.(.-' -5.), y=^.(2'-e,), ^E') 



x', y, i!, being the coordinates of the near ray. And if we eliminate «, j8, by these equations, 

 from the general equation (N) 



which represents all the pencils of the system, we find for the general equation of thin pencils. 



^=^c-^> 



[42.] These equations (E'), (P) include the whole theory o thin pencils. As a first appli- 

 cation of them, let us suppose that we are looking at a luminous point, by means of any combi- 

 nation of mirrors ; the rays that enter the eye will not in general diverge from any one focus, and 

 therefore will not be bounded by a cone, but by a pencil of another shape, which I shall call 

 the Bounding Pencil of Vision, and the properties of which I am now going to investigate. 



Suppose for this purpose, that the optic axis coincides with that given ray of the reflected 

 system which we have taken for the axis of (z), and let (J) represent the distance of the eye from 

 the mirror ; the circumference of the pupil will have for equations 



2 = 3, x' -|- J/* = «•, 



