Best Pavilion Angle

Since no pavilion angle will reflect all incident light back to the crown, it is necessary to satisfy the next best criterion. For this, Tolkowsky chose that light returned to the table should be refracted so as to produce the optimum combination of intensity and dispersion. He says that these are related to the to the external angle as follows:

Intensity (I) is proportional to the cosine .......... I = cosf
Dispersion (D) is proportional to the sine .......... D = sinf

The product of these factors is I•D = sinf•cosf =1/2•sin2f.

...which is extreme when 2f = ±90^o ... thus when... f = ±45^o

The corresponding internal angle q of this ray in diamond is ...

sinq = sinf / RI = sin(±45^o) / 2.417 ... thus... q = ±17^o

The average of all rays incident to the table is f = 0^o, for which ... q = 0^o substituting q = 0^o and q = ±17^o into Eq.(1c) gives ...

0^o±17^o = 4(45^o-P) ... thus... P = 45^o±4.25^o = 40.75^o, 49.25^o

If P=40.75^o

qmin = -16.32^o, ..... qmax = +33.32^o, ..... qmax-qmin = 49.64^o

fmin = -42.8^o, ..... fmax = +90^o, ..... fmax-fmin = 132.8^o

If P=49.25^o

qmin = -24.82^o, ..... qmax = +7.82^o, ..... qmax-qmin = 32.64^o

fmin = -90^o, ..... fmax = +19.2^o, ..... fmax-fmin = 109.2^o

Tolkowsky's reasons for selecting the lower value (40.75^o) are not theoretically sound. There are certain advantages to the higher value (49.25^o). but there are more reasons to select the lower value:

1. 1. The stone transmits more light (fmax-fmin = 132.8^o vs. 109.2^o),
   2. It can be viewed through a greater angle without seeing through it (f=42.8^o vs 19.2^o)