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Thus the wave KE, while advancing toward the meeting place becomes abc, whereof the part ab is made by the evolute bC, a portion of the curve ENC, while the end C remains attached; and the part bc by the evolute of the portion bE while the end E remains attached.
At present the federal government is building a railroad which will tap some of the resources of the region. Enc. Brit., "Alaska." By 1908, the year of the presidential election, an influential portion of the Republican members of Congress, particularly in the Senate, were bitterly opposed to President Roosevelt.
Consequently the same wave becomes def, then ghk, and finally CY, from whence it subsequently spreads without any fold, but always along curved lines which are evolutes of the curve ENC, increased by some straight line at the end C.
Now to say what these waves become after the rays have begun to cross one another: it is that from thence they fold back and are composed of two contiguous parts, one being a curve formed as evolute of the curve ENC in one sense, and the other as evolute of the same curve in the opposite sense.
And the surface EK which all those waves will touch, will be the continuation of the wave AD in the sphere at the moment when the piece D has reached E. Now the line EK is not an arc of a circle, but is a curved line formed as the evolute of another curve ENC, which touches all the rays HL, GM, FO, etc., that are the refractions of the parallel rays, if we imagine laid over the convexity ENC a thread which in unwinding describes at its end E the said curve EK. For, supposing that this curve has been thus described, we will show that the said waves formed from the centres F, G, H, etc., will all touch it.
It is certain that the curve EK and all the others described by the evolution of the curve ENC, with different lengths of thread, will cut all the rays HL, GM, FO, etc., at right angles, and in such wise that the parts of them intercepted between two such curves will all be equal; for this follows from what has been demonstrated in our treatise de Motu Pendulorum.
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