326 OF THE MOTIONS OF FLUIDS. 



This proposition, though not belonging to the motions 

 of fluids, is inserted here to complete the analogy between 

 the height of a liquid, the modulus of elasticity of an elastic 

 medium, and the modulus of tension of a vibrating chord. 

 The force, impelling any small portion of the chord towards 

 the quiescent position, or axis, is obviously expressed by the 

 diagonal of the elementary parallelogram, formed by its 

 extreme tangents, that is the Hne intercepted between the 

 intersection of those tangents and a line equal and parallel 

 to the second drawn from the extremity of the first, or in 

 other words, by the second fluxion of the ordinate, when 

 the tangent represents the first fluxion of the axis, the 

 curve being always supposed infinitely near to the axis, 

 and in general the force will be to the tension as the 

 second difference AAy to the first difference Ax : but the 

 tension is to the weight of the element a^ as M to ax, con- 

 sequently the tension of ax is — g, and the accelerative 



force — - • — cizz — ^Mqzz--^ Mat, which we may maker:/' 



AX AX-^ AX^ ^ dx^ ^ J J 



d^V 

 — __^, and we shall have i;= s/(gM), as vzz s/{gy) in ar- 



tide 378 ; and the velocity will be that which is due to 

 half the height M. 



The reflection at the extremities of the chord may 

 be represented by delineating the initial figure, and re- 

 peating it in an inverted position below the absciss: then 

 taking, in the absciss, each way, a distance propor- 

 tional to the time ; and the half sum of the correspond- 

 ing ordinates will indicate the place of the point at the 

 expiration of that time. The chord will thus represent a 

 portion of the surface of a liquid agitated by a series of 



