FUNDAMENTAL IDEA OF WAVE MECHANICS 307 rays). It is surprising that a general principle as important as Fermat's relates directly to these mathematical guide lines, and not to the wave surfaces, and one might be inclined for this reason to consider it a mere mathematical curiosity. Far from it. It becomes properly understandable only from the point of view of wave theory and ceases to be a divine miracle. From the wave point of view, the so-called curvature of the light ray is far more readily understandable as a swerving of the wave surface, which must obviously oc- cur when neighbouring parts of a wave surface advance at different speeds, in exactly the same manner as a company of soldiers marching forward will carry out the order"right incline" by the men taking steps ofvarying lengths the right-wing man the smallest, and the left-wing man the longest In at- Aa ospheric refraction of radiation for example( Fig. 2 )the section of wave arface Ww must necessarily swerve to the right towards Ww because its left half is located in slightly higher, thinner air and thus advances more apidly than the right part at lower point. (In passing, I wish to refer to one point at which the Snellius view fails. A horizontally emitted light ray should remain horizontal because the refraction index does not vary in the horizon- al direction. In truth, a horizontal ray curves more strongly than any other which is an obvious consequence of the theory of a swerving wave front. On detailed examination the Fermat principle is found to be completely tantamount to the trivial and obvious statement that-given local distribution of light velocities- the wave front must swerve in the manner indicated cannot prove this here, but shall attempt to make it plausible. I would again ask you to visualize a rank of soldiers marching forward. To ensure that the line remains dressed, let the men be connected by a long rod which each holds firmly in his hand. No orders as to direction are given; the only ord is: let each man march or run as fast as he can. If the nature of the ground varies slowly from place to place, it will be now the right wing, now the left that advances more quickly, and changes in direction will occur spon- taneously. After some time has elapsed, it will be seen that the entire path travelled is not rectilinear, but somehow curved. That this curved path is exactly that by which the destination attained at any moment could be at- tained most rapidly according to the nature of the terrain, is at least quite plausible, since each of the men did his best. It will also be seen that the swerv ing also occurs invariably in the direction in which the terrain is worse, so that it will come to look in the end as if the men had intentionally"by- passed"a place where they would advance slowly The Fermat principle thus appears to be the trivial quintessence of the waveFUNDAMENTAL IDEA OF WAVE MECHANIC S 307 rays). It is surprising that a general principle as important as Fermat’s relates directly to these mathematical guide lines, and not to the wave surfaces, and one might be inclined for this reason to consider it a mere mathematical curiosity. Far from it. It becomes properly understandable only from the point of view of wave theory and ceases to be a divine miracle. From the wave point of view, the so-called curvature of the light ray is far more readily understandable as a swerving of the wave surface, which must obviously oc￾cur when neighbouring parts of a wave surface advance at different speeds; in exactly the same manner as a company of soldiers marching forward will carry out the order "right incline" by the men taking steps ofvarying lengths, the right-wing man the smallest, and the left-wing man the longest. In at￾mospheric refraction of radiation for example (Fig. 2) the section of wave surface WW must necessarily swerve to the right towards W1W1 because its left half is located in slightly higher, thinner air and thus advances more rapidly than the right part at lower point. (In passing, I wish to refer to one point at which the Snellius’ view fails. A horizontally emitted light ray should remain horizontal because the refraction index does not vary in the horizon￾tal direction. In truth, a horizontal ray curves more strongly than any other, which is an obvious consequence of the theory of a swerving wave front.) On detailed examination the Fermat principle is found to be completely tantamount to the trivial and obvious statement that - given local distribution of light velocities - the wave front must swerve in the manner indicated. I cannot prove this here, but shall attempt to make it plausible. I would again ask you to visualize a rank of soldiers marching forward. To ensure that the line remains dressed, let the men be connected by a long rod which each holds firmly in his hand. No orders as to direction are given; the only order is: let each man march or run as fast as he can. If the nature of the ground varies slowly from place to place, it will be now the right wing, now the left that advances more quickly, and changes in direction will occur spon￾taneously. After some time has elapsed, it will be seen that the entire path travelled is not rectilinear, but somehow curved. That this curved path is exactly that by which the destination attained at any moment could be at￾tained most rapidly according to the nature of the terrain, is at least quite plausible, since each of the men did his best. It will also be seen that the swerv￾ing also occurs invariably in the direction in which the terrain is worse, so that it will come to look in the end as if the men had intentionally "by￾passed" a place where they would advance slowly. The Fermat principle thus appears to be the trivial quintessence of the wave