Small Vibratory 3Iotions of Elastic Fluids. 277 



as retaining the same sign. It follows from the preceding reason- 

 ing, that the motions of the particles are always such as result 

 either from a motion of propagation in a single direction, or 

 from two simultaneous motions of propagation in opposite direc- 

 tions. 



3. In order to find out the nature of the functions F and / 

 conceive two propagations exactly alike, to produce the same 

 motions of the particles in the same order, but to have contrary 

 directions. There must be one point at least where the motions 

 are constantly the same at the same instant, in virtue of the 

 two propagations, but in opposite directions. Consequently v 

 must = at this point, whatever be t. If I be its distance from 

 the origin of a-, F {I - at) +/{l + at) = o. Describe two curves, 

 the equation of one of which is y ^ F {I - z), and of the other 

 J/=/(^ + 2)- Then because F(l—z) = -f{l + z), for the same value 

 of 2 the ordinates are equal with opposite signs. Hence, if one 

 of the curves be transferred to the opposite side of the axis, the 

 two will coincide. This transfer is made by changing the sign 

 of y ; consequently -f{l + z) =F{1+ z), and F {I- z) = F {l + z). It 

 appears thus, that the curves which F and / embrace must all 

 satisfy the above condition, the meaning of which is, that a point 

 may be taken in the axis of z, such that the ordinates at equal 

 distances from it on each side are equal. Again, if two curves 

 exactly alike, and possessing this proi3erty, move in opposite 

 directions, there must of necessity be a position in which they 

 coincide. Hence, f may be changed into F, and for all positions 

 of the curves, v = F [x - at) + F {x + at). Let the distance of a 

 point at which t; = o whatever be t, from the origin, be = I', 

 a quantity related to Z in a manner which will presently be de- 

 termined, and not necessarily the same as I, because the origin of 

 X is arbitrary. Hence, F{1' - at}= — F{1' + at); a condition which 

 all the curves given by F and / must satisfy, and which shews that 



