FIRST ORDINARY MEETING. I 



Professor Young then read a paper on the " Sokitions of 

 Equations of the Fifth Degree." The object of the paper 

 was, in the first place, to determine the criterion of the 

 solubility of the quintic equation ; and next, assuming the 

 conditions of solubihty to exist, to solve the equation. 



A short discussion followed, in which Mr. Livingston and 

 Prof. Galbraith took part. 



Prof J. Loudon also read a paper entitled : — 



GEOMETRIC A.L METHODS CHIEFLY IN THE THEORY 

 OF THICK LENSES. 

 1. In cases of reflection or refraction at a spherical surface, or a 

 combination of sphei^ical surfaces, or lenses, if F, F' be the pi-imary 

 and secondary principal foci of the surface, lens, or combination, and 

 (P, P'), (R, R') pairs of conjugate points, it is known that 



■^+^,= 1 (1) 



P P 

 where/ = RF, ]) = ^P,/' = R'F', p' = R'P' ; and where the posi- 

 tive direction from R for / and p is opposite to, whilst that from R' 

 for /' and p' is the same as, the direction of the incident pencil. 

 Now since the relation (1) expresses the condition that the line 



--(-—= 1 passes through the point (/",/'), it follows that if the 

 P V 



coincident lines FRR'F', FRR'F' be separated so that R on the x or 

 object-axis coincides with R' on the y or image-axis, the line joining 

 P on the former to P' on the latter will always pass through the 

 fixed point {f,f'). Hence we dei'ive a geometrical method for 

 determining the point conjugate to any given one. 



The points R, R' from which distances are measured, it is to be 

 observed, are any two conjugate points, such, for example, as the 

 principal points, or nodal points ; and they may in particular cases 

 coincide when they are self-conjugate. 



It is proposed in the following paper to employ the method 

 indicated chiefly in discussing certain propositions in the theory of 

 thick lenses. 



I. 



'2. In the case of refraction at a single spherical surface 



■^ + =^ = 1 



J) p i 



