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from the surface, till, at no great distance, it is at rest in space. According to the undulatory theory, the direction in which a heavenly body is seen is normal to the fronts of the waves which have emanated from it, and which have reached the neighbourhood of the observer, the aether near him being supposed to be at rest relatively to him. If the aether in space were at rest, the front of a wave of light at any instant being given, its front at any future time could be found by the method explained in Airy’s Tracts. If the ether were in motion, and the velocity of propagation of light were infinitely small, the wave’s front would be displaced as a surface of particles of the aether. Neither of these suppositions is however true, for the aether moves while light is propagated through it. In the following investigation I suppose that the displacements of a wave’s front in an elementary portion of time due to the two causes just considered take place independently. Let w, v, w be the resolved parts along the rectangular axes of jt, y 2 = C + ......(I*) C being a constant, t the time, and £ a small quantity, a function of x, y and t. Since «, v, w and £ are of the order of the aberration, their squares and products may be neglected. Denoting by a, /3, 7 the angles which the normal to the wave’s front at the point (#,y, z) makes with the axes, we have, to the first order of approximation, dt » d£ and if we take a length V dt along this normal, the co-ordi-nates of its extremity will be If the aether were at rest, the locus of these extremities would be the wave’s front at the time t + dt, but since it is in motion, the co-ordinates of those extremities must be further increased by udt> vdt, wdt. Denoting then by ^,y, z | which corresponds to the point (x’ *? = x+(u-V&)dt, y= d =± z + (to + V) d t; and eliminating ,v zf — {w + V)dt=C + V/ ## +/{/ - (» - vg) it, y - (»- vg) i/, <},or, expanding, neglecting dt z = C + Vt + 1'+^w + V)dt. . . . (3.) But from the definition of £ it follows that the equation to the wave’s front at the time t + dt will be got from (1.) by putting t 4- dt for t z=C + V* + ?+ (V + ifydt. . . . (4.) Comparing the identical equations (3.) and (4.), we have ## <*K77 = This equation gives £= J”wdt: but in the small term £ we may replaceylwrZrf by ^ J*mdz\ this comes to taking the approximate value of z given by the equation z = C + V/, instead of t z=C + V^-f—J*wdz. Combining the value of £ just found with equations (2*), we get, to a first approximation, 7r 1 rdtxi J 7T 1 /*dw J : ## * 2 ~ vJ dx * P 2 “ VJ dy ’ ^equations which might very easily be proved directly in a more geometrical manner. If random values are assigned to a |