384 J. H. ROHRS, ESQ. ON THE 



A x = - .85 A cosv/c#.24.2 — t. 

 b 2a 



The chain being supposed started from instantaneous rest. 



(The other type is \/cg.8.2 — J. 



4. A far better way, however, to get at the types of vibration is the following, which was 



d 2 

 pointed out to me by Professor Stokes. Since it is clear that if we were to eliminate — (A/ ), 



from the infinite number of equations (5), we should, (with the condition J, - A^ + A 3 - &c. = 0), 



have the same types in every term A lt A.-., &c. ; let qs/cg — be a type common to all the 

 terms, substituting for 



d? 

 And eliminating — M between the first and each succeeding set of equations, we have, 

 dv 



A l (l-q t ) + A t {#(?-f)mQ, 



A 1 (l-q 2 )-A 3 {5 2 (5 2 -q s ) = 0, 



&c. &c. - 0. 



And Ax - A 2 + A 3 - &c. = ; 



1 1 1 



1 - q* 3 2 (3 2 - q*) ' 5 s (5 2 - q 2 ) 



+ &c. = . . . . (7). 



5. There is another yet simpler mode of determining q, for which I am also indebted to 



a hint of Professor Stokes', who advised me to attack equation (4) at once, without the use of 



dV 

 series, which (when it is not required to have — expressed in a series of a particular form,) 



ds 



is by far the shortest plan. 



d 2 idV\ 2s d 2 .. /<?P 



„ d 2 idV\ 2s d* „ (d?V\ . 



For d* ml + * ; *? M * ' cg \m may be wntten 



d 2 (dV 2s \ d 2 IdV 2s . \ 



dV 28 , — 



••• -j- + -j M = h sin (p i/eg t) sin (ps + a). 



<PV 

 Now -^-3 — = ; .-.0 = 0; 



and = 0, in order that u. may = ; 



