Theory o/Mr. B. Hamilton's String-Organ. 101 



which is the complete equation of the problem. The calculation 

 of \ in the general case presents great difficulty ; and before a 

 comparison of theory with experiment could be effected, it would 

 be necessary to measure the length, weight, and tension of the 

 string, the distance of the point of attachment from one end, 

 the note of the reed alone (A), and the load (X ). The last ele- 

 ment cannot be directly ascertained by any means with which I 

 am acquainted. 



The following are particular cases in which equation (7) is 

 satisfied. 



I. If n, r are integers and — =b or =/ — b, the left-hand side 



of (7) vanishes ; equating the right-hand side to zero, we must 



, , r\ 7 XI 



nave also =- = /, or - = -. 

 2 2 r 



Now ^ is the length of a segment of the string which would 



give the note actually sounded. Hence in this case the note 

 sounded is a harmonic of the string alone. And the case arises 

 when the point of attachment is a node. 



It is obviously true that the string can vibrate in one of its 

 ordinary harmonics if the point of attachment remains at rest ; 

 for in this case neither the load nor the elastic force of the reed 

 comes into play. But the case is not a solution of the problem. 



II. Again, if A=\, or the note sounded be that of the reed 

 alone, the right-hand side of (7) is to be equated to zero, and 

 the note sounded is a harmonic of the string alone, as before. 

 The case is that in which the reed and string would, if inde- 

 pendent, vibrate simultaneously. For suppose the attachment 

 severed; the reed will go on speaking its own note and the 

 string its harmonic ; and as these are the same note, the motion 

 will go on as if the attachment continued to subsist. This is 

 obviously a possible solution of the problem. 



III. If X be indefinitely diminished, we have ultimately the 

 right-hand member of (7)= 0, as before, and the string can 

 sound any of its ordinary harmonics. 



This is the case in which the effect of the peculiar arrange- 

 ment of the reed is insufficient to modify sensibly the normal 

 properties of the string. It is possible that this case may be 

 realized by the employment of a thick and heavy metallic wire. 



IV. On the assumption of certain relations between the ele- 

 ments, the formula (7) reduces in complexity. The simplest 

 assumption that can be made is, that the point of attachment is 

 at the middle of the string. According to experiment no discon- 

 tinuity in the nature of the results arises at thi3 position ; con- 



