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VI. 



ON THE SYMMETRICAL OPTICAL INSTRUMENT. 

 By H. C. FLUMMEK, M.A 



Read May 27, 1918. Published January 31, 1919. 



1. In the first of a series of three papers 1 on geometrical optics, published in 

 1905, the late Professor Schwarzschild has treated the errors of an optical 

 instrument on the basis of Hamilton's characteristic function, or Eikonal, 

 as it has been called by Brims. His method assumes the results of the 

 Gaussian first approximation ; but little more than a re-arrangement of the 

 work is required to give a simple and self-contained theory of the errors 

 to the next order of approximation. There may be some advantage in 

 reproducing the theory in this form. 



The whole rests on the single principle of Fermat, that the optical path 

 of a ray between two points P , P x is a minimum, or more exactly, that the 

 effect of a first-order displacement of intermediate points on the length of 

 path is of the second order. From this it follows that the rays from P form 

 a normal congruence in isotropic media, the normal surfaces being defined by 



2> = E (x, y, 9, x a , y ot z ) = const., 



where s is the length of ray in a medium with index fi from the initial point 

 (x , y , z ) to the end point (x, y, z). The effect of a displacement of I\ is 

 measured by the projection on the normal to the eikonal surface passing 

 through P x , and is therefore 



dE = im (liBxt + mjS^i + ihBzi) 



where (/,, m u «i) are the direction cosines of the ray at P,. If an initial 



1 Astr. Mitth. der K. Sternwarte zn Gottingen, ix-xi. Abhandl. der K. Ges. der 

 Wiss, zu Gottingen, Math.-phys. Kl., Neue Fulge, B. iv. 



