Vi'oi.' Stoke^'nExaminaiion of this possible effe'ct &f 



^■uThils equation is satisfied by ' it^iu n^ivr, vn^.ti-j >o iiiw Jl 

 -Y.-j g—^^»!04^t-'i ifsrf* .m'fc'^ 3^afi« ^''fsll 



where A' is an arbitrary constant^ real or imaginary, and ?w'^|»? 

 ^Jw9^}fs^l gr^^i^jg^ry, poii^tants coAi;^^f\^f (^.li^y ;tbfti9(i\iftt^9^ , , 



.b'intuJdo ii-jodbEfl a I^^ih, >» '■ {n' + q)n'^ n(]/ .3 /;imhiiini odi vjm 

 .o!cf!^.ii'j^ifi m\ ^ ru8 -?: ~ k{{l+a^)u' + qy dnVrrir.) aHt ?/oA •) 

 L',f\\iy Hi( i>..f,r tiw! r^, . . ,, i,..i •J triiauD biiuo^ 



If we suppose m' wholly imagmarj-^ tbe ionniuse will reter to 



an infinite series of waveSj tbe expressions for s, &c. involving as 

 under tbe circular functions sine and cosine. In this case our 

 formulae would raabe known tbe manner in which tbe motion 

 alters with the time. If we suppose n' wholly imaginary, tbe 

 motion will be periodic as regards tbe time. In this case we 

 must not suppose tbe fluid unlimited, but bounded in one direc- 

 tion by a vibrating plane which keeps up the motion. I shall 

 select the second case for consideration, inasmuch as it is analo- 

 gous to that of the vibrations propagated along a long tube from 

 a sonorous body at one end of it, and accordingly Avill bear on 

 tbe experiments by which ]\I. Biot proved that the velocity of 

 propagation of sound in air is independent of tbe pitch. 



Let the origin be situated at the vibrating plane, and let. lis 

 consider the motion of tbe fluid situated at the side of cT positive. 

 Let ??f be what m! becomes when ?i' is replaced by \^ — In. Tbe 

 equation (9.) furnishes two values of m, corresponding to two 

 series of waves, which travel, one in the positive, and the other 

 in tbe negative direction. Of coursfe Vre Elrfe only cbheeyned With 

 the former. We get from (9.) -"""''- ^in - i- y " """"i"' 



y(P|erftp'^.„.)f{j !,,; ; ^ ,!,!i.;i ..|iii:it ti) 'ijinr.rh r>[di8n3« on ad !i ' 

 ni'-'C lno rio/r-' •j d'Si / iioi>-?>^1n*"g^'> r'tiV ib -jib 'jilt no Offob iljuv 

 Jfno -ynr.-^, V/t\(l +a/3)'?i^+9V':b!q->o '::fti'rnb 7i6 ;)fff vU 



Choosing that root of rnr which coiTcsponds to. waves ti'aveilittg 

 in the positive direction, we get from (10.) ihhm! i. ■ 



Substituting in (8.), introducing another function got by 

 changing tbe sign of -/ — 1 and taking a new arbitrary constant, 

 changing the arbitrary constants so as to get rid of tbe imaginary 

 quantities, and altering the origin of the time so as to get rid of 

 one of the circular functions, we:g^i'i niJ amjJ jilf ^}\. .yiuJiii 

 - sinii/' r ' I' I-- iTiimni vd boqobvob iii'xl 

 -«9-,q9rft7d nnfr^.A^M;^/..!-.!.'' COS {»?/-/. COS ./r.a?),^ ^.-,„ ^ 413.) 



