1876.] the Principle of Reciprocity to Acoustics. 121 



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 is greater 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 im- 

 probable, 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 understand*. 



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 at a 

 short distance apart, to be acting simultaneously. By calling the tv^-o 

 sources opposite, it is meant that they are to be at any moment in op- 

 posite 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, how- 

 ever, 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 ^sAth. 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 direc- 

 tions 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 velocity of the medium 

 at B resolved in the direction B Q due to the source at A is the same as 

 the velocity at A resolved in the direction A P due to the source at B. 

 If the waves observed at A and B are sensibly plane, and if the axes 

 A P, B Q are equally inclined to the waves received, we may, in the above 

 statement, replace " velocities " by " pressures," but not otherwise. 



Suppose, now, that equal double sources face each other, so that ths 

 common axis is A B, and let us examine the effect of interposmg a screen 

 near to A. By the reciprocal theorem, whether there be a screen or not, 

 the velocity at A in direction A B due to B is equal to the velocity at B 



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

 one of Mr. F. Galtoii's whistles, which gives notes near the limits of ordinary hearing. 

 VOL. XXV. K 



