100 Mr. R. H. M. Bosanquet on the Mathematical 

 Either (3) or (4) gives for oc=.b 



~ = — « 2 m 2 sin mb sinm(l— b) (A cos amt + B sin amt). 



And taking the difference of the values of ~- given by the two 



equations when a?=b, we have 



du 

 A-^=— w2sinm/(Acos«m/ + Bsin amt). 



Hence, in order to satisfy (1), we must have 



/xa^m 2 sin mb sin m{l—b)=Tm sin ml+a sin m(l—b) sin mb, 

 or 



{jxa^m 1 — ct) sin mb sin m (l—b)= Tm sin m/. . (5) 



This equation determines m, while A and B remain arbitrary 

 unless the initial circumstances of the motion are given. 



Put fi = p\ , where X is the length of the string, the weight 

 of which is equal to the load. 



Put - = «V 2 , so that av=— — . where /is the periodic time of 

 fj, t 



the reed with the load vibrating alone. This combination will 



be referred to as " reed alone/' 



Then a = a?v' 2 . p\ , and equation (5) becomes 



(w 2 — j/ 2 )X sin mb sin m(l—b) =m sin ml. . . (6) 



Now a is the velocity of transmission along the string. Let 

 t be the periodic time of the vibration actually sounded, X the 

 corresponding complete wave-length on the string (i. e. A,= 

 twice the length of a single segment). 



Then, from the form of (3), (4), 



_37T_27T 



ar X " 



Similarly, if A be the wave-length on the string of the note of 

 the reed alone, 



2tt 

 V= -A- 



This transformation is convenient ; but it must be remembered 

 that the notes denoted by X and A are those which would be 



given by single segments of lengths -, - respectively. Making 



the substitutions above indicated, the equation (6) becomes 



27r(A 2 — X 2 )X sin — — sin — K — — = A 2 X sin — -, . . (7) 



