FIRST ORDINARY MEETING. 9 
6. The following proposition which is employed by Helmholtz 
(wide Optique Physiologique, p. 72), 1 have modified by changing 
his notation and applying the rule of signs (§ 1), in order to exhibit 
the result of the elimination in a symmetrical form. 
Let there be any number of spherical refracting surfaces whose 
principal foci are (F), F’,), (F,, F’:), &c., and which cut the common 
principal axis in A, B,C,... Let (R,, R,), (R,, R.)... be pairs 
of conjugate points, such that R,F, = d,, R,F’,=—d',,... In like 
manner let (P,, P,), (P;, P,) . . . be any other set of conjugate points, 
such that R,P, = p,, R,P, = 7',,... Then by § 4 
ke ati a =l, 
ert 
&; + ds = ale &e 
Py 2 
Also by the rule of signs (§ 1) we have p, = —p’,, pp = — p's, . 
Hence, on eliminating these quantities, the position of P,,, the point 
conjugate to P, with reference to the system, is determined from an 
equation of the form 
f cS De PE eee mere 
where f= R,F, /’ = R,F’, F, F’ being the principal foci of the 
system. 
The values of f for 2, 3, 4. . . refractions are, respectively, 
9 
dq, dds ase MEG eds nae 85 
weed dd,+dd,+d a, ddds+d dd,+d dd,+d dd, — 
an a Ji 
dis, ds. wa 
d', did’) d',d',d'. 
dtd, dd,+d'd,+ dd’, dydod; +. . ye 
7. The construction of § 5 (Fig. 2) applies to equation (3), and 
from the figure we at once deduce, as in §§ 3, 4, the general relations 
; a d’ 
dd a ’ BD) =F D’ 
The latter, it may be observed, also follows from (3), since R,, R,, are 
any conjugate points. 
and the corresponding values of are 
eae 
