h'eaa^ 



dax* , Idax' 



+ ir+; 



ADDITIONS 



beax' 



85 



2 . . gJ'^ " ■ ■ I' ■ 3 . . ei" ■ ■ ■ P ' i . .yP 

 + . , . , for the fourth fluxion of this expression, divided by 

 b, is of the same form with the expression itself ; and the 

 number of terms allows it to fulfil all the conditions that 

 may be required. In both the cases here proposed, the co- 

 efficients d and e vanish, because the second and third flux- 

 ions are initially evanescent, and the equation becomes 



y=a+ 



l-a.T* b'ax'^ b'ax" i '""'"^i '"'"^ 



b^cax^ 



'a . .' gP 





In the first case, when 

 b 



x—l,OT-'^l, y—0, and i'^0, whence 1-) — 



b* P 



Q . . 8 2 . . 12 



+ ...+C + 



b 



be 



2 . . 5 



b-c 

 2 . . 9 



. 4 



b'c 



■*-... =0, and 1 1 — 



+ <•+ 



13 

 be 



t'c 



iV 



1+- 



2 . . 8 

 b 



. 12 

 b' 



11 2 . .4 



+ . . . =0 ; therefore — cz:: 



— + • • . 



, and z: 



2 . . 12 



i + - 



2 . . 5 



b . V 



-+■- 



2 . . 9 



2 . . 7 2 



2 . 



:+.■ 



13 



1 + 



b b' 



2 . 4 a . 



r+a, 



i^ 



Hence, by mul- 



12 



tiplying the numerator of each fraction by the denominator 

 of the other, and arranging the products according to the 



1 4 



Powersoft, we obtain the equation i b+ i'— 



3 .4 3 . .8 



— — i'+ . . .=0, which has an infinite number of roots; 



3. . 12 * 



the first two being ir:i2.3623, and ^=489.4. In a simi- 

 lar manner we obtain, for the second case, making the se- 

 cond fluxion of y, and either its third fluxion, or the area, 



16 



here call d : but the weight of the particle x' is -- x', and 



A 



ddh "y . . „, 



the force is to that of gravity as — r-;-"^ '* '° unity. Now 



y _ ba.x" , . . . , 



; for, when x is evanescent, the subsequent terms 



XX ll 



are inconsiderable in comparison with this, and the force is 

 , the space to be described being a ; and if the spate 



12i' 



12^4 



became , and the force equal to that of gravity, the 



bddk 



vibration would be performed in the same time : this is 



therefore the length of the synchronous pendulum ; that is, 



•9707 1* 

 for the fundamental sound, in the first case j., , and in 



be thesecond.023976-^Tr' 



A pendulum, of which the length is ^^j, feet. 



makes 



and 



ddh 

 / h 39.13\ .^ . . 



I . I vibrations m a second, 



\.9707 1.2 / ' 



— ^( .— ^ )=:n double vibrations, such as 



2/'^ \.9707 12 / 



/nll\i 

 A=l.igo7( — ) • And in the same manner, for a rod 



vanish when x:=2, the equation b -f. 



3.4 3 . .8 ^3..ia 



<!4 



Ji' i'+ . . . zzo : and of this the first two roots 



3 . . 16 



are i=:500.5 and iiZ3803. From these values of /', those 

 of c may be readily found ; and for each value after the first, 

 the rod has an additional quiescent point. 



In order to determine the time of vibration, we must com- 

 pare the force acting on a particle x/ at the end of the rod P"^""' *= "^*"*' ^°"^ '^'" ^i'^ S^at as in the prism, 



jj^ and the time will be increased in the subduplicate ratio, or 



with its weight. The force is ^^ (32 1), a being equal ^ j ,„ .ggg. jf ^ cylinder be compared with a prism of 

 ^j. the same length and weight, its vibrations will be less fre- 



to \x', r to _. (194), and b being the depth, which we may q„j„,i„ t^e ratio of 300 to 307, or nwriy of 43 to 44. 



are considered in the estimation of musical sounds. Hence 



/nll\t 

 loosely supported at two points, A=:.0294l \'t] ' 



When the rod is loosely fixed at both ends, the figure 



coiacides with the harmonic curve, (398. SchoLium), and 



lai' 

 the length of the equivalent pendulum is , ,^. , c being 



3.1416, and c'.or b, 97.41. 



If a prismatic bar supported at the extremities, be depres- 

 sed by a weight equal to a portion of itself of which the 



length is gl, the depression being e, A will be —rr , and 



4CM1C 



nH* g 



when *="t73T)'''=8.5— , e being expressed infeet. The 



weight under which the bar may begin to bend, (sas) will 

 be equal to that of a portion of which the length is. 



The stiffness of a cylinder being to that of its circumscrib- 

 ing prism as three times its mass to four times that of the 



