PROBLEM OF THE VIBRATIONS OF STRINGS. 33 



winch is called into action by the motion of the string : for it is mani- 

 fest that, when the string is drawn aside from the straight line into 

 which it is stretched, there arises an additional tension, which aids in 

 drawino- it back to the straight line as soon as it is let go. Hooke 

 (On Spring, 1678) determined the law of this additional tension, 

 which he expressed in his noted formula, " Ut tensio sic vis," the 

 Force is as the Tension ; or rather, to express his meaning more 

 clearly, the Force of tension is as the Extension, or, in a string, as the 

 increase of length. But, in reality, this principle, which is important 

 in many acoustical problems, is, in the one now before us, unimport- 

 ant; the force which urges the string towards the straight line, 

 depends, with such small extensions as we have now to consider, 

 not on the extension, but on the curvature ; and the power of 

 treating the mathematical difficulty of curvature, and its mechanical 

 consequences, was what was requisite for the solution of this pro- 

 blem. 



The problem, in its proper aspect, was first attacked and mastered 

 by Brook Taylor, an English mathematician of the school of Newton, 

 by whom the solution was published in 1 715, in his Methodus lucre 

 mentorum. Taylor's solution was indeed imperfect, for it only pointed 

 out a form and a mode of vibration, with which the string might move 

 consistently with the laws of mechanics ; not the mode in which it 

 must move, supposing its form to be any whatever. It showed that 

 the curve might be of the nature of that which is called the companion 

 to the cycloid ; and, on the supposition of the curve of the string being 

 of this form, the calculation confirmed the previously established laws 

 by which the tone, or the time of vibration, had been discovered to 

 depend -on the 'length, tension, and bulk of the string. The mathe 

 matical incompleteness of Taylor's reasoning must not prevent us from 

 looking upon his solution of the problem as the most important stop 

 in the progress of this part of the subject : for the difficulty of apply- 

 ing mechanical principles to the question being once overcome, the 

 extension and correction of the application was sure to be undertaken 

 by succeeding mathematicians; and, accordingly, this soon happened. 

 We may add, moreover, that the subsequent and more general solu- 

 tions require to be considered, with reference to Taylor's, in order to 

 apprehend distinctly their import; and further, that it was almost 

 evident to a mathematician, even before the general solution had ap- 

 peared, that the dependence of the time of vibration on the length 

 ind tension, would be the same in the general case as in the Taylo- 



