the displacement of a particle is found to be D' D ar ar3• The import of this expression will best be seen by assuming a form for the function D. Suppose the same form of function which is found convenient in the Undulatory Theory of Light for a series of similar waves in continued succession, and to which (as we shall hereafter shew) all systems of recurring similar waves may be referred. The value of this function, after going through various changes with continued increase in the value of r, returns, when r is increased by λ, to the same value as before; and as the characteristic of a series of similar recurring waves is that at the end of a certain spatial interval, which we call "the length of a wave," the state of disturbance is the same as at the beginning of that interval, it follows that λ is the length of a wave. The same recurrence of value is produced with unaltered r if we increase t by; which shews that the wave advances through a space equal to its length, so as to leave a succeeding wave exactly in the same place λ in which the preceding wave was, in the time 2. Now, since D=b.sin 2 (at − r), or=b. sin 2v, - sin (2+0), arcos e -2/(1+0). sin (2+0). αλη The angle increases from 0 (when r=0) to (when r is oo). Thus we find that, The displacement of the particles is expressed by a modified wave, in which the maximum of backwardsand-forwards disturbance is not the same at all distances from the center of divergence, but varies more rapidly than the reciprocal of distance from that center. For great distances, however, it is proportional to the reciprocal of distance. The progress of the modified wave is not uniform; 2πυ for to the quantity there is attached 0, or to v there λθ 2πT' λ is attached a quantity increasing positively but more rapidly at first than at last. Conceive this united This shews that at the time t the multiple at in v is connected with r', a quantity smaller than r. To ascertain the spatial interval of waves at a given time 7, we 2π must changer' by such a quantity that r' will be λ changed by 27, that is, r' will be changed by λ, or λθ r- will be changed by λ, or r will be changed by 2π Hence the spatial interval of the waves is rendered rather larger by this term; the interval in time being, at any given point, necessarily unaltered (as determined H only by the interval of impulses at the origin), and the velocity of the waves is a little increased. But the whole gain in space travelled over by a wave is 51. Divergent wave of three dimensions, with oscillation of the center of divergence in the direction of z. Assume F to be Rz, R being a function of t and r only, where r = √(x2 + y3 + z3). Then, by a process exactly similar to that of Art. 46, =2. dx dr'dx R R x2z dR y3z + z + dx2 = dr3 · x2 + y2+z2 + dr' (x2 + y2 + z2)* ' (by the expansion of last Article). Similarly, d'F dy' = ďR y'z dR x2z + z |