Page:Optics.djvu/82

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58

When is between and , is between and This may easily be seen from the geometrical construction, (Fig. 77.) or it may be shown from the formula: for

which shows that is greater or less than according as it is greater or less than

When comes to , coincides with it.

By differentiating the equation we find

which shows that the distance is at a maximum in the space between and when , or

for when is at a maximum and

If we place on the other side of (Fig. 78.) or make negative, we shall have

whence we collect that as long as is negative and increasing: that when or is at , is infinite, and that afterwards it becomes positive, or that goes to the other side of

Obs. It will probably have occurred to the reader, that by placing within the denser medium, we have virtually passed from the first case to the fourth, with the only difference that the places of and are inverted. I have, however, purposely placed in all possible positions, in order to illustrate the connexion between the