188 PHILOSOPHICAL TRANSACTIONS. [ANNO 1782. 



fig. 14, which, on account of its passing to the focus unrefracted, maybe called 

 the axis of the pencil, can never be found in the axis of the telescope eo, except 

 at the focus f, where d and p meet. That ray however op, parallel to bg, 

 which falls obliquely on the axis of the telescope eo, will continue to pass along 

 it after refraction, and for that reason it may be called the relative axis of the 

 pencil. This will appear, by considering that the particle of light, which at any 

 moment is refracted at the vertex o of the spherical surface, is found by hypo- 

 thesis in the axis a second time, when it meets the contemporary light at the 

 focus. But since the motion both of the axis and of the particle is uniform and 

 rectilinear, the former cannot be found in the latter at 2 different times, without 

 being found in it continually during the whole interval. In like manner, a part 

 of every other ray from the star, which successively falls on the vertex, must 

 move relatively along the axis after refraction : and thus a constant succession of 

 these particles constitute a visual refracted ray, whose relative path must always 

 be in the axis oe. 



All that has been shown concerning the telescope already considered, will re- 

 ceive still further illustration, by tracing the motion of this particular refracted 

 ray till it arrives at the focus. This way of viewing the subject will also render 

 the reasoning more general, and make it apply to telescopes when the dense 

 fluid within is supposed to be confined by object-glasses of any figure. But in 

 order to this, it will be convenient to premise, and briefly to demonstrate, what 

 shall afterwards be referred to by the name of 



Prop. A. — If any very small body or particle of light, as it moves uniformly 

 in the absolute path sb, fig. l6, has passed relatively along a part of the line cd, 

 which advances equably and parallel to itself in the direction dk ; and if at any 

 instant the absolute path of the particle be changed into any other, as br ; then 

 it will still pass relatively along the moving line, provided its velocity now be to 

 its former velocity, as the sine of the angle dbf to the sine of the angle dbr ; 

 these being the angles which the moving line bd makes with bf and br, the ab- 

 solute path or direction of the particle in the two cases. 



The construction of this figure is so simple, that it is unnecessary formally to 

 point it out. Since, by hypothesis, the velocity of the particle along br is to its 

 former along bf, as the sine fz to the sine rt ; or, on account of similar tri- 

 angles, as df to ir, and, on account of parallels, as df to dw, it follows, that 

 the time of its describing br now, is to the time of formerly describing its equal 

 bf, as dw to df. But the line bd advancing with a uniform motion, the time 

 of its arriving at w is to the time of its arriving at f, also as dw to df. There- 

 fore, when the particle arrives at r, the point d of the moving line will have ar- 

 rived at w, and wrp will be its position. Hence the particle at that moment 

 must be found in the intersection r of this line, with its absolute path br. In 



