A prism is found to give a minimum deviation of 51, The same prism gives a deviation of 62°48' for lose values of the angles of incidences 40° 6' and 82°42', Determine (a) Refracting angle of prism. (b) Refractive index of the material of the prism.

Kwa mkengeuko wowote $\delta$, ikiwa $i_1$ na $i_2$ ni pembe mbili za mwangaza zinazotoa mkengeuko huo, basi moja inaweza kuchukuliwa kama pembe ya mwangaza ($i$) na nyingine kama pembe ya kutoka ($e$).
Hivyo, tunachukua $i = 40^\circ$ na $e = 82^\circ 42'$.
Kanuni ya mkengeuko ni:

$$\delta = i + e - A$$

$$A = 40^\circ + 82^\circ 42' - 62^\circ 48'$$

Kwanza, jumlisha pembe za mwangaza na kutoka:

$$40^\circ + 82^\circ 42' = 122^\circ 42'$$

Kisha toa mkengeuko:

$$A = 122^\circ 42' - 62^\circ 48'$$

Ili kutoa, tunahitaji kukopa $1^\circ$ kutoka $122^\circ$ na kuibadilisha kuwa $60'$:

$$A = 121^\circ (42' + 60') - 62^\circ 48'$$ $$A = 121^\circ 102' - 62^\circ 48'$$ $$A = (121 - 62)^\circ (102 - 48)'$$ $$A = 59^\circ 54'$$

Pembe ya prism ni $\boxed{59^\circ 54'}$.

$$n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$

Tunajua $A = 59^\circ 54'$ na $\delta_m = 51^\circ$.

$$n = \frac{\sin(55^\circ 27')}{\sin(29^\circ 57')}$$

Kubadilisha dakika kuwa desimali kwa hesabu:
$55^\circ 27' = 55 + \frac{27}{60} = 55.45^\circ$
$29^\circ 57' = 29 + \frac{57}{60} = 29.95^\circ$

$$n = \frac{\sin(55.45^\circ)}{\sin(29.95^\circ)}$$ $$n \approx \frac{0.82360}{0.49924}$$ $$n \approx 1.6497$$

Fahirisi ya mtawanyiko ni $\boxed{1.650}$ (kwa sehemu tatu za desimali).