726 PHILOSOPHICAL TRANSACTIONS. [anNO 1703, 



sines, and i for the common sine of incidence ; for we shall have as before v^i 

 to r — i in a given ratio ; which call that of i to q. And from these two, we 

 have ^^ = y^ X y But p is always nearly equal to q ; in the refractions 

 from glass, and from water into air, their difference is less than -j-^ part of either; 

 we may therefore put the ratio ~ . equal to ^-^ which is the first theorem. 



And thence, if one difference r — i become equal to r — ?, the other differences 

 V — I, &c. will be respectively equal to i; — i, &c. and the same set of differences 

 may be made to serve for several media, provided the sines of incidence are taken 

 in their due proportion. Thus when red rays pass from glass into the air, we 

 have I : r :: 50 : 77 and r — i : i :: 27 : 50, and when they pass from water into 

 air / : r — i :: 3:1, and therefore, as we are to make r — i every where equal to 

 r — I, we get, ex aequo, ? : i :: 81 : 50, as Sir Isaac Newton finds it. 



Remark 4. But to explain this matter a little further, and obviate some diffi- 

 culties concerning it, I shall add the following examples. 



The refractive powers being marked as above, let red rays fall from glass into 

 air at the angle of incidence 20°, the angle of refraction will be 3 1'^ 47'. Again, 

 let them fall from water into air at an angle of 34^^ l', making their angle of re- 

 fraction 48° 5'. And the difference of the sines of 31° 47' and 20° will be pre- 

 cisely equal to the difference of the sines of 48° 5' and 34" l'. At the same 

 angles of incidence 20° and 34° l', let the violet rays fall from glass and water 

 into air, and the angle of refraction from the glass will be 32° 14'-i. nearly, and 

 that from the water will be 48° 38 nearly. And the difference of the sines of 

 32° 14'-!^ and 20° will be equal to the difference of the sines of 48° 38' and 34° l' 

 within .000488, or less than ^Vo-o part. We see likewise that the red and 

 violet rays diverged from the glass medium at an angle of 27'4^ ; but from the 

 water at an angle of 33' ; making the difference of divergence in this example 5^', 

 that is i of the whole divergence of the red and violet rays when refracted from 

 glass into air, at incid. 20°. Whence it appears, that though the differences of 

 the sines above specified, or the excesses in Sir Isaac's theorem, may in refractions 

 from different media into the same rarer medium, be made equal, it does by no 

 means follow, that the divergences of the several sorts of rays (or if you chuse 

 to call it their dispersion) will be the same in the 2 refractions ; for Sir Isaac's ex- 

 cesses 27, 27-L, &c. are the excesses of sines ; not of angles, as some opticians seem 

 to have misapprehended. 



/Vgain, let an unrefracted pencil of light fall from common glass into the air, 

 fig. 7, at the incidence 39", and the angles of refraction will be. 



Of the violet 79° 2' 2" 



Of the red 75 43 55 



And their difference. ... 318 7 is the divergence of 



