17t) Prof. Ohallis on the I lydrody mimical Theory 



Thus the circumstances of spontaneous uncompounded motion 

 relative to an axis are in all respects determined, and with any 

 required degree of approximation. 



8. This result is of primary importance relatively to the 

 mathematical theory of musical vibrations, as will appear from 

 the following considerations. In the case of aerial vibrations, 

 for which within the usual compass of musical notes X is large 

 and consequently, as e\ 2 is constant, e is very small, each of 

 the factors f, y, h, &c. will be nearly unity, and we may sup- 

 pose ~- to vanish. Then for the velocity in the direction of 

 z we have 



d<f . m 2 A ra 3 B . Q y s 



—J—-=zm sin ql cos 2qt — sin ogc + &c. 



az J a * cr 



Now as this expression has been obtained prior to any suppo- 

 sition as to the mode of disturbing the air, it is to be taken 

 into account in any case of motion due to arbitrary disturb- 

 ance. Representing therefore by ¥ the value of the principal 

 variable when it applies to motion produced under given arbi- 

 trary circumstances, let us suppose, in order to fulfil the above- 

 mentioned condition, that 



-=— = 2/x . m sin ^ 9 cos lq 1 % %~~ sm fyi fe + & c - 



cr^ l. ^' ^* 



+ 2 2 . m! sin g^f cos 2^ 2 ? / g-sin ^^f + &c. 



+ &c, 



the symbol S signifying the sum of any number of such ex- 

 pressions as that contained within the brackets. For these 

 expressions m may be supposed to have the respective arbitrary 

 values m ly m 2 , m s , &c. ; and c in f to have the corresponding 



27T 



arbitrary values c 1? c 2 , c 3 , &c, whilst o , 1 , or — -, has the same 



value for all the expressions embraced by the first sum (Si). 

 Similarly for those embraced by the second sum (2 2 ),the arbi- 

 trary quantities are m\, m' 2 , m / 3 , &c, and c\, c' 2 , c f z , and q 2 , 



27J- 



or — , is the same in all. If now account be taken of the arbi- 



A, 2 



trary quantities that are thus at disposal, it may readily be 

 shown that the above equation may be put under the form 



