of the Principle of Reciprocity to Acoustics. 303 



reflect the vibrations in the direction of the flame, the effect is 

 again produced with the same apparent force as at first. 



These results prove conclusively that the reed does not behave 

 as the simple source of theory, even approximately. When the 

 screen is close (about 2 inches distant) the more powerful vibrations 

 issuing along the axis of the instrument impinge directly upon 

 the screen, are reflected back, and take no further part in the ex- 

 periment. The only vibrations which have a chance of reaching 

 the flame, after diffraction round the screen, are the compara- 

 tively feeble ones which issue nearly at right angles with the 

 axis. On the other hand, when the screen is close to the flame, 

 the efficient vibrations are those which issue at a small angle with 

 the axis, and are therefore much more powerful. Under these 

 circumstances it is not surprising that the flame is affected in the 

 latter case and not in the former. 



The concentration of sound in the direction of the axis isgreater 

 than would have been anticipated, and is to be explained by the 

 very short wave-length corresponding to the pitch of the reed. If, 

 as is not improbable, the overtones of the note given by the 

 reed are the most efficient part of the sound, the wave-length 

 will be still shorter and the concentration more easy to under- 

 stand*. 



The reciprocal theorem in its generalized form is not restricted 

 to simple sources, from which (in the absence of obstacles) sound 

 would issue alike in all directions ; and the statement for double 

 sources will throw light on the subject of this note. A double 

 source may be thus defined : — Conceive two equal and opposite 

 simple sources, situated a short distance apart, to be acting 

 simultaneously. By calling the two sources opposite, it is meant 

 that they are to be at any moment in opposite phases. At a 

 moderate distance the effects of the two sources are antagonistic 

 and may be made to neutralize one another to any extent by 

 diminishing the distance between the sources. If, however, at 

 the same time that we diminish, the interval, we augment the 

 intensity of the single sources, the effect may be kept constant. 

 Pushing this idea to its limit, when the intensity becomes infinite 

 and the interval vanishes, we arrive at the conception of a double 

 source having an axis of symmetry coincident with the line joining 

 the single sources of which it is composed. In an open space the 

 effect of a double source is the same as that communicated to the 

 air by the vibration of a solid sphere whose centre is situated at 

 the double point and whose line of vibration coincides with the 

 axis, and the intensity of sound in directions inclined to the axis 

 varies as the square of the cosine of the obliquity. 



The statement of the reciprocal theorem with respect to double 

 sources is then as follows : — If there be equal double sources at 

 two points A and B, having axes A P, B Q respectively, then the 



* July 13. — I have lately observed that the flame in question is extremely 

 sensitive to one of Mr. F. Galton's whistles, which gives notes near the limits of 

 ordinary hearing. 



