107 



(e) being the radius of tlie pupil ; the rays of the bounding pencil of vision pass through this cir- 

 cumference, and therefore satisfy the condition 



and eliminating «, ^, from this, by means of (D'), we find the following equation for the bound- 

 ing pencil of vision, 



^-A\^ = eK (G') 



It is evident, from this equation, that every section of the pencil by a plane perpendicular to 

 the optic axis, that is, to the given ray, is a little ellipse, having its centre on that ray, and its 

 semiaxes situated in the tangent planes to the two developable pencils, that is in the planes of 

 {x,z), {y, z). Denoting these semiaxes by (a), (5), we have 



a = ±e.^:=^, b=.±e.L=^; (H') 



these semiaxes become equal, that is, the little elliptic section becomes circular, first when 



z' =: 3, a=b = e, 

 that is at the eye itself, and secondly when 



that is, at a distance from the eye equal to the harmonic mean between the distances of the eye 

 from the two foci of that reflected ray, which coincides with the optic axis. It may also be 

 proved, that when the eye is beyond the two foci, the radius of this harmonic section, (which is 

 to the radius of the pupil as the semi-interval between the two foci is to the distance of the eye 

 from the middle point between them,) is less than the seraiaxis major of any of the elliptic sec- 

 tions, that is, than the extreme aberration of the visual says at any other distance from the eye; 

 so that, in this case, we may consider the centre of the harmonic section as the visible image of 

 the luminous point, seen by the given combination of mirrors ; observing however that the appa- 

 rent diitante of the luminous point will depend on other circumstances of brightness, distinct- 

 ness and magnitude, as it does in the case of direct vision with the naked eye. 



[43.] One of the principal properties of thin pencils, is that the area of a perpendicular sec- 

 tion of such a pencil is always proportional to the product of its distances from the two foci of 

 the given ray. We may verify this theorem, in the case of the bounding pencil of vision, by 

 means of the formulae (H') for the semiaxes of the little elliptic section; in general if we repre- 

 sent by £ the area of the section of any given thin pencil, corresponding to any given value of 

 (r'), we shall have by (E') 



2.2 =/ (T/d^; — My) = (/ — e , )(2' — «,)./ (^d» — »dfi), (10 



and the definite integral /(/3c;« — ttdfi), depending only on the relation between », /S, is constant 



