taEORYOF THE MOTION OP ROCKETS. ^ ggl 



by expanding the fraction and taking the fluent of eacli. 



a m . c /' 



term have, for the loff. the series — x ( < + 



^ a m — c t am \ 



■- 4- r— i — i + -; — 3 — 3 + -r— r~4 + ^^ \ Hence th^ 



2am 3a m* 4a^»i^ 5a*m* J 



above fluxional expression becomes x zz. x ( < <* + 



b c /t"^ ct^ c"- t* 



whose fluent is x = — X ( tt + ^ + -Ti — i i + 



am V 2 o am \2 a m 



; — ; + 1 — 1 + &c J — S f^, which wants no cor^ 



SOa^m^ ^ 30a*m* / ^ 



h c 

 rection : therefore in the case where t — a', x z:z — X 



a m 

 /a* , ca* , c'-a^ , c^ a^ ^ c''"^ c \ 



VT+el^ + Ii^^ + i^l^r^ + i^^+^V'-" ^ = 



0*5"; the space through which the roclvct ascends during 



the time of its burning. 



Hence retaining the numbers In the example above for 



^, , . , ,, , 6774-075 X 3 X 160 



the velocity, we shall have x = X 



•' 2 X 448 



f 4. l^Q 16 0^ 160 ' 160* 



V 3 X 448 6 X 448 * 10 X 448 ' 15 X 448* 



4- kc\ — 144 rr 362-8-96875 X 1-14622279 (the sum of 



the series to 6 terms) — 144 — 4159*606684 — 144 = 

 4015'6o6684 feet, the space the rocket ascends through 

 during the 3 seconds it is on fire. 



The fluent of 6 f. hyp. loff. g t t\ mieht 



•^' ° a m — ct 



indeed have been obtained \ylthont a series ; for b t'. hyp. ''l'*'' "^ 'j'**^'- 



■' ' minuiiT the 



log:. zz b t . hyp. log. am — b t . iiyp. log;, [a m 



rr-ct) the fluent of the former part of which is evidently 

 b t. hyp. log. a m ; and the fluent of / . hyp. log. [am — c t) 



Another me- 



