PROFESSOR AT HEIDELBERG 193 



optical length is proportional to the time in which the light 

 traverses the length of the ray, and equal to the distance which 

 the light would have traversed in the same time in empty 

 space. The law of refraction of light -rays may accordingly 

 be expressed by saying that the optic length of the ray between 

 given points in the first and last medium must be a limiting 

 value (maximal or minimal) when the refracting media are 

 limited by surfaces of continuous curvature. In following up 

 the analogy with the potential function, Helmholtz finds that 

 if the rays have started from any point, and are broken by an 

 indefinite number of surfaces of continuous curvature, they 

 will after the last refraction be perpendicular to that curved 

 surface for the points of which collectively the optical length 

 of the ray is of constant value. This surface contains all points 

 at which the same phase of ether vibration occurs, and is 

 accordingly a wave surface. After laying down this theorem he 

 applies the known properties of the normal and of the curva- 

 ture of a surface to the determination of the course of the rays 

 in an infinitesimally thin bundle of rays. But this further gives 

 the laws of the refraction of bundles of rays in prisms. Helm- 

 holtz finds that an infinitesimally thin bundle of homocentric 

 rays, starting from a point at infinite distance, will only remain 

 homocentric after its passage through a prism if it has passed 

 through at an angle of minimal deviation, i. e. if it is in a plane 

 perpendicular to the refracting edge, and makes an equal 

 angle with both surfaces of the prism. Since a luminous 

 point can only form a clear image when the refracted light 

 is homocentric, the accuracy of the image of a line of light is 

 evidently not affected by deviations of the rays, provided 

 they lie in the direction of its image. From this he arrives at 

 the images of luminous objects, when these consist of vertical 

 bright lines of different, monochromatic lights ; and is able to 

 determine the brilliancy of the spectrum, and to prove that 

 its brightness, apart from loss by reflection and absorption, 

 is directly proportional to the brightness of the spectral colours 

 involved, and the apparent breadth of the slit inversely pro- 

 portional to the apparent length of the part of the spectrum 

 that is in question. 



Meantime Helmholtz had determined, while still engaged 

 on his Physiological Optics, to write a similar work on Sensa- 



