Mr. J, J. Waterston on the Theory of Sound. 485 



Thus if the motion of a particle 'is oscillatory, like a com- 

 plete cycloidal pendulum, the required sequence of force demands 

 the above specific sequence of change in the distance of the par- 

 ticles. Again, if the motion of each particle is oscillatory, the 

 I'equired sequence in its velocity (viz. that it should vaiy as x 

 the sine) demands also a specific sequence of change in the di- 

 stance of the particles ; and this sequence is precisely the same 

 as what is required by the sequence- of 'force. 



To obtain a clear idea of the proof of this (which is a problem 

 of pure mathematics), we may supposeVith the same radius / 

 another semicircle to be described, placed also in the line of the 

 pulse, and removed to the distance A, from the preceding semi- 

 circle. Let a third also be drawn, removed the same distance \ 

 from the second. We have further to suppose these semicircles 

 divided into as many parts (aaj, a^Oc^, a^a^, &c.; bb^, b^b^, 6363, &c.; 

 cc„ CjCg, CgCg, &c.), beginning at where the line of pulse inter- 

 sects them as \ enters into L (or — number of parts). The 

 length of each of these parts or steps is thus ^ 2l7r=s (being in- 

 finitesimal with regard to \). 



Having made this construction, we have next to consider that 

 the motion of each particle to be oscillatory must be such that, 

 at the instant when particle A has traversed the versed sine of 

 aa„, the particle B (next in advance of A) being one step behind 

 in its motion, has traversed only the versed sine of bbn-\, and 

 particle C the versed sine of cc„_2. If B had traversed as many 

 steps as A, the distance X that separates them would not alter; 

 but since it is a step behind, AB is at this point less than \ by the 

 difference between vers ««„ and vers bbn-i, or vers ««„— vers aan-\, 

 which equals s . sin aOn-i- In the same way, C being a step 

 behind B, their distance is less than Xhy s . sin«fl„_^. Thus 



we have BC — BA = s (sin a«„— sin afl„_i) =s cos aa„ j. [Here s, 



being an absolute magnitude, has to be divided by the absolute 

 radius / to represent the differential of arc] At the beginning 

 of the vibration n=l, and cosafl„= radius; hence with B at 

 the initial point b, C at c_i (a step back on the returning half of 

 the previous oscillation), and A at «j (the points on the circle 

 being supposed projected on the diameter), the difference 



BC-BA=i'=4<^;. 



This initial amount determines the accelerative force acting at 

 the beginning of the motion of each particle, which is obtained 

 by comparing it with the rccii)rocal of \, which represents the 



