OF DEFLECTIVE FORCES. l61 



Scholium 5. A more general analytical solution of 

 the problem may be obtained by making s—b sinf + c sin 

 (et-i-h) whence v=.—fj'dtzz'^jn^ (6—1) sin t-\-c sin {et 



-f A) >d^=«< (6—1) cos f + - cos (etr^h) J +f, since d cos 



(et-\'h)—^sm{et-\-h)edt; and s=:fvdtzznk {b—l) sin * 



+ ~ sin (et + h) I -hit + kzzb &m t + c sm(et-\-h)\ whence 

 ee J 



nc o 



n(b — 1)=:6, — i=c, i=0 and ^=0; consequently w=£ — r- 

 ee o — 1, 



1 n 



and 6= -, — = 1, and e=: V«, A and c remaining alto- 



w — 1 ee 



gether undetermined. We may, therefore, accommodate 

 this expression to any relative values of the supposed vi- 

 brations, or of the forces belonging to them, and to any 

 conditions of motion or rest in the initial state of the moving 

 body. Thus, if we suppose it initially at rest, so that 5=zO 

 and vzrO when <ziO, the length n being given, we have 

 5=6 sin t-\-c sin (ef + ^)zzO, and consequently Az:0, and 



C C C J_ 



«=;:» (6 — 1) cos *+— cos e*=6 + -=0, and -= — 6= 



e e e n—l 



whence cz: r^r- — > ^^^ we have szz -, + - — sm 



n — 1 1 — n n — 1 n — 1 



s/nt. 



295. Theorem. " B.'' If the resistance 

 be simply proportional to the velocity, a pen- 

 dulum with a vibrating point of suspension 

 may perform regular vibrations, isochronous 

 with those of the point of suspension, provided 



M 



