INSTRUMENTS ILLUSTRATING KINEMATICS. 67 



more rapid. The " harmonic " tones of a string, which are always 

 heard along with the fundamental tone, are a particular case of 

 these constituents. The theorem was given by Fourier in con- 

 nection with the flow of heat, but- its applications are innumerable, 

 and extend over the whole range of physical science. 



The laws of combination of harmonic motions have been illus- 

 trated by some ingenious apparatus of Messrs. Tisley and Spiller, 

 and by a machine invented by Mr. Donkin ; but the most im- 

 portant practical application of these laws is to be found in Sir W. 

 Thomson's Tidal Clock, and in a more elaborate machine which 

 draws curves predicting the height of the tide at a given port for 

 all times of the day and night with as much accuracy as can be 

 obtained by direct observation. One special combination is 

 worthy of notice. The union of a vertical vibration with a hori- 

 zontal one of half the period gives rise to that figure of 8 which 

 M. Marey has observed by his beautiful methods in the motion of 

 the tip of a bird's or insect's wing. 

 . ,. The motion of the sun and moon relative to the earth 



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Motion. was at rst Described by a combination of circular mo- 

 tions ; and this was the immortal achievement of the Greek astrono- 

 mers Hipparchus and Ptolemy. Indeed, in so far as these motions 

 are periodic, it follows from Fourier's theorem mentioned above 

 that this mode of description is mathematically sufficient to repre- 

 sent them ; and astronomical tables are to this day calculated by a 

 method which practically comes to the same thing. But this 

 representation is not the simplest that can be found ; it requires 

 theoretically an infinite number of component motions, and gives 

 no information about the way in which these are connected with 

 one another. We owe to Kepler the accurate and complete 

 description of planetary or elliptic motion. His investigations 

 applied in the first instance to the orbit of the planet Mars about 

 the sun, but it was found true of the orbits of all planets about the 

 sun, and of the moon about the earth. The path of the moving 

 body in each of these motions, is an ellipse, or oval shadow of a 



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