390 



Transactions of the Society. 



the first, second and fourth in air, the third in glass. If we write 

 A /3 2 /3 3 and /3 4 for the diameters of the several images, u x u 2 u 3 , 

 and w 4 for the divergence angles of the image-forming beams, and n x 

 % and w 4 for the refractive indices of the media in which 



n 



they are severally formed, we may write Helmholtz' proposition 

 symbolically thus : 



iii sin u x /?! = n 2 sin u 2 /3 2 = % sin % /3 3 

 = w 4 sin Ui /3 4 — &c. ad infinitum. 



It will of course be understood that the divergence angle u is 

 the angle formed with the optical axis by the ray which touches 

 the edge of the aperture. 



Of this proposition Helmholtz gives, as I have said, two proofs. 

 The first is borrowed from and credited to Lagrange. Helmholtz 

 accompanies his reproduction of it with a criticism pointing out that 



Fig. 81. 



it is imperfect, inasmuch as it applies only to divergence angles * 

 of infinitesimally small magnitudes. 



Helmholtz therefore, with handsome acknowledgments to 

 Lagrange, propounds a new and very elegant proof of his own. 



Divested of its mathematical expression and thrown into the 

 form of an imaginary experiment, Helmholtz' proof may be 

 explained as follows. 



Let e in the diagram (fig. 81) be a board on which are mounted 

 a number — any number — of electric lamps arranged in squares, 

 quincunx, or otherwise, so as to secure their even distribution over 

 the surface of the board. At A let there be a circular aperture 

 filled by a lens, and let rj be another board fitted with a number of 



* In the unrevised proof which was circulated before the meeting on the 18th 

 March I inadvertently wrote " images" for " divergence angles" in this place. 



