Small Vibratory Motions of Elastic Fluids. 295 



by taking' m positive and negative from o up to its limiting 

 value, we get a number of stiaight lines, all of which belong 

 to possible motions. Hence the motion may be such as is shewn 

 by two straight lines ac, cb, (Fig. 6.) inclined differently to the 

 axis and meeting at c. The whole of the preceding reasoning 

 is applicable mutatis mutandis to vibrating chords, and this line 

 may consequently represent that form of the chord, which, being 

 practically possible, led Euler to start the idea of discontinuotis 

 functions to account for it theoretically. D'Alembert, unwilling 

 to admit into analysis any thing so singular, thought that the 

 theory was inapplicable to cases of this kind. By taking an 

 unlimited number of small portions of the straight lines included 

 in the equation y = Trmx, it would be possible to form a segment 

 of any continuous curve by joining them together ; — for instance, 

 a segment of a circle ; — and the motion indicated by such a line 

 would be possible, provided always the ordinates be small, and 

 at every point the tangent be inclined at a small angle to the 

 axi.s. 



It thus appears that the form of the functions F and f, is 

 not necessarily that which I have called primary: they may 

 take any other forms whatever, subject to the limitatibns above- 

 mentioned. These forms are given when the particular mode 

 of the disturbance is given ; or, since the disturbance may always 

 be supposed to be made by the motion of a diaphragm, the 

 form of the function is given when the particular motion of the 

 diaphragm is given. It is very necessary to know what is the 

 primary form of these functions, in all eases in which the motion 

 to be considered, results from the action of the parts of the 

 fluid on one another. All that has been said from Art. 7. will 

 serve to shew that it is by reason of the discontinuity of 

 the motion, that the functions are susceptible of other forms, 

 which correspond to an action of a different kind, as for 



