FIRST ORDINARY MEETING. 7 
Professor Young then read a paper on the “ Solutions 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 solubility 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 :— : 
GEOMETRICAL METHODS CHIEFLY IN THE THEORY 
OF THICK LENSES. 
1. In cases of reflection or refraction at a spherical surface, er a 
combination of spherical surfaces, or lenses, if F, F’ be the primary 
and secondary principal foci of the surface, lens, or combination, and 
(P, P’), (R, R’) pairs of conjugate points, it is known that 
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where f= RE, p = RP, f = RF’, p' = RP’; and where the posi- 
tive direction from R for f and p is opposite to, whilst that from R’ 
for f’ and p’ is the same as, the direction of the incident pencil. 
Now since the relation (1) expresses the condition that the line 
~+4 4 —1 passes through the point (f°), it follows that if the 
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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,/’). Hence we derive 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 
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