∴
m
Δ
(
Δ
′
−
r
)
=
Δ
′
(
Δ
−
r
)
;
{\displaystyle \therefore m\Delta (\Delta '-r)=\Delta '(\Delta -r);}
∴
Δ
′
=
m
Δ
r
m
−
1
¯
Δ
+
r
,
o
r
1
Δ
′
=
m
−
1
m
r
+
1
m
Δ
.
{\displaystyle \therefore \Delta '={\frac {m\Delta r}{{\overline {m-1}}\Delta +r}},\quad \mathrm {or} \quad {\frac {1}{\Delta '}}={\frac {m-1}{mr}}+{\frac {1}{m\Delta }}.}
(Case 1, 4.)
m
=
Δ
+
r
Δ
⋅
Δ
′
Δ
′
+
r
;
{\displaystyle m={\frac {\Delta +r}{\Delta }}\cdot {\frac {\Delta '}{\Delta '+r}};}
∴
m
Δ
(
Δ
′
+
r
)
=
Δ
′
(
Δ
+
r
)
;
{\displaystyle \therefore m\Delta (\Delta '+r)=\Delta '(\Delta +r);}
∴
Δ
′
=
m
Δ
r
−
m
−
1
¯
Δ
+
r
,
o
r
1
Δ
′
=
−
m
−
1
m
r
+
1
m
Δ
.
{\displaystyle \therefore \Delta '={\frac {m\Delta r}{-{\overline {m-1}}\Delta +r}},\quad \mathrm {or} \quad {\frac {1}{\Delta '}}=-{\frac {m-1}{mr}}+{\frac {1}{m\Delta }}.}
77. There is another expression sometimes used, in which the distances are measured from the centre, (Fig. 70 .)
Let
E
Q
=
q
,
E
q
=
q
′
,
{\displaystyle EQ=q,\quad Eq=q',}
m
=
Q
E
A
Q
⋅
A
q
E
q
=
q
r
−
q
⋅
r
−
q
′
q
′
;
{\displaystyle m={\frac {QE}{AQ}}\cdot {\frac {Aq}{Eq}}={\frac {q}{r-q}}\cdot {\frac {r-q'}{q'}};}
∴
q
(
r
−
q
′
)
=
m
q
′
(
r
−
q
)
;
{\displaystyle \therefore q(r-q')=mq'(r-q);}
∴
q
′
=
q
r
m
r
−
m
−
1
q
¯
,
o
r
1
q
′
=
−
m
−
1
r
+
m
q
.
{\displaystyle \therefore q'={\frac {qr}{mr-{\overline {m-1q}}}},\quad \mathrm {or} \quad {\frac {1}{q'}}=-{\frac {m-1}{r}}+{\frac {m}{q}}.}
78. It will be observed, that we have taken
m
{\displaystyle m}
to represent the ratio of the sines of incidence and refraction in all cases, whether the passage of the light be into a denser or a rarer medium; if we chuse that
m
{\displaystyle m}
should always represent the ratio of the sines of incidence and refraction out of the rarer into the denser, we must, in Cases 3 and 4, put
1
m
{\displaystyle {\frac {1}{m}}}
for
m
{\displaystyle m}
.
Then
Δ
∓
r
Δ
⋅
Δ
′
Δ
′
∓
r
=
1
m
,
{\displaystyle {\frac {\Delta \mp r}{\Delta }}\cdot {\frac {\Delta '}{\Delta '\mp r}}={\frac {1}{m}},}
and
1
Δ
′
=
∓
m
−
1
r
+
m
Δ
.
{\displaystyle {\frac {1}{\Delta '}}=\mp {\frac {m-1}{r}}+{\frac {m}{\Delta }}.}