234 Prof. Silvanus P. Thompson on 



relative velocity in any medium is called its " velocity- 

 constant"; it is the reciprocal of the index of refraction. 

 The velocity-constant, for mean (yellow) light, for water 

 is about 0*75; that of crown glass 0*65 ; that of flint glass 

 from 0'61 to 0'56, according to its density. 



3. Method of Reckoning Curvature. 



The Newtonian definition of curvature as the reciprocal 

 of the radius has a special significance in the present method 

 of treating optics: for some of the most important of lens 

 and mirror formulae consist simply of terms which are 

 reciprocals of lengths, that is to say of terms which are 

 curvatures. The more modern definition of curvature as 

 rate of change of angle per unit length of the curve (Thomson 

 and Tait's c Natural Philosophy,' ii. p. 5) is equivalent to New- 

 Eton's ; for if in going along an arc of length 8s, the direction 

 changes by an amount 80, the curvature is 80/8s. But the 

 angle 80 = 8s/r, where r is the radius of curvature; hence the 

 curvature = Bs/rBs = 1/r. 



There is, however, another way of measuring curvature, 

 which, though correct only as a first approximation, is 

 eminently useful in considering optical problems. This way 

 consists in measuring the bulge of the arc subtended by a 

 chord of given length. 



Consider a circular arc AP, having as its centre. Across 

 this arc draw a chord PP X of any desired length. The 

 diameter A B bisects it at right Fig. 1. 



angles in M. The short line M A 

 measures the depth of the curve 

 from arc to chord. If the radius 

 is taken as unity the line M A is 

 the versed-sine of the angle sub- 

 tended at B by the whole chord, or 

 is the versed-sine of the semi-angle 



subtended at the centre. In Continental works it is fre- 

 quent to use the name sagitta for the length of this line 

 M A ; and as this term is preferable to versed-sine, and 

 can be used generally irrespective of the size of radius, it 

 is here adopted. The proposition is that, for a given chord, 

 the sagitta is (to a first degree of approximation) propor- 

 tional to the curvature. For it follows from the con- 

 struction that 



MA.MB= (PM) 2 ; 

 assuming PM as unity, 



MA = 



MB ~ 2r- AM" 



