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IN RESISTING MEDIUMS, $$$
fcribed between the two pofitions. For v (in the defcent of
rtC
:, theva-
the body) being, univerfallu, equal to —._
J v aa 4- 2C-U
lue of c, exprefiing the celerity at the vertex A, will be had
from that equation, and comes out rr ■ ———•
v aa — 2z?2Q
whence alfo z (zz d X hyp. log. -) rr d X hyp. log.
v
a
rr — \d X hyp. log. 1
2vvQ
From
Vaa —- 2vzQ aa
which, the celerity at A being known, the reft is obvious.
But, in the afcending part of the curve EA, both z and Q
muft be confidered as negative, or wrote with contrary figns :
and then, from the foregoing equations, we fliall alfo get v rr
ac av , t r i
■:, c zz ■■; . , and — zr |d X hyp.
*/ aa — 2ccQ
Vaa F 2vvQ
log. 1
2ccQ
aa
— \dx hyp. log. 1 F
2vvQ
aa
; and, con
fequently, z rr — \d X hyp. log. 1
2ccQ
aa
zz \d X hyp.
2vvQ v
log. 1 -| rr d X hyp. log. - : anfvvering in this cafe.
tilt C
It ftill remains to take fome notice of the values of a: and y
(in order to have the form, as well as the length, of the
curve). Thefe, indeed, are not fo eafy to bring out as that of
a, given above ; nor can they be exhibited in a general manner, either by circular arcs or logarithms (that I have been
able to difcover) ; but may, however, be approximated to any
required degree of exactnefs, as will appear from what
follows.
Since z (rr AB) is found rr \d X hyp. log. ~a
by taking the fluxion thereof, we get k zz
czdw^ I 4- wz
aa
ccdQ
aa F 2ccQ
au F 2c2 Q
(becaufe Q zz w^ 1 F ^2)« Therefore j> (=:
| Title | Doctrine and application of fluxions. |
| Alternative Title | The doctrine and application of fluxions containing (besides what is common on the subject) a number of new improvements in the theory, and the solutions of a variety of new and very interesting problems in different branches of the mathematics... To which is prefixed an account of his life. The whole revised and carefully corrected by William Davis. |
| Reference Title | Simpson, Thomas, 1805, The doctrine and application of fluxions. |
| Creator |
Simpson, Thomas, 1710-1761. Davis, William, 1771-1807 . |
| Subject | Calculus. |
| Publisher | London. |
| DateOriginal | 1805 |
| Format | JP2 |
| Extent | 31 cm. |
| Identifier | 138 |
| Call Number | QA303.S45 1805 |
| Language | English |
| Collection | History of Mathematics |
| Rights | http://www.lindahall.org/imagerepro/ |
| Data contributor | Linda Hall Library, LHL Digital Collections. |
| Type | Image |
| Title | Page 335. |
| Format | tiff |
| Identifier | 1138_363 |
| Relation-Is part of | Is part of: The doctrine and application of fluxions containing (besides what is common on the subject) a number of new improvements in the theory, and the solutions of a variety of new and very interesting problems in different branches of the mathematics... To which is prefixed an account of his life. The whole revised and carefully corrected by William Davis. |
| Rights | http://www.lindahall.org/imagerepro/ |
| Type | Image |
| OCR transcript | IN RESISTING MEDIUMS, $$$ fcribed between the two pofitions. For v (in the defcent of rtC :, theva- the body) being, univerfallu, equal to —._ J v aa 4- 2C-U lue of c, exprefiing the celerity at the vertex A, will be had from that equation, and comes out rr ■ ———• v aa — 2z?2Q whence alfo z (zz d X hyp. log. -) rr d X hyp. log. v a rr — \d X hyp. log. 1 2vvQ From Vaa —- 2vzQ aa which, the celerity at A being known, the reft is obvious. But, in the afcending part of the curve EA, both z and Q muft be confidered as negative, or wrote with contrary figns : and then, from the foregoing equations, we fliall alfo get v rr ac av , t r i ■:, c zz ■■; . , and — zr |d X hyp. */ aa — 2ccQ Vaa F 2vvQ log. 1 2ccQ aa — \dx hyp. log. 1 F 2vvQ aa ; and, con fequently, z rr — \d X hyp. log. 1 2ccQ aa zz \d X hyp. 2vvQ v log. 1 -| rr d X hyp. log. - : anfvvering in this cafe. tilt C It ftill remains to take fome notice of the values of a: and y (in order to have the form, as well as the length, of the curve). Thefe, indeed, are not fo eafy to bring out as that of a, given above ; nor can they be exhibited in a general manner, either by circular arcs or logarithms (that I have been able to difcover) ; but may, however, be approximated to any required degree of exactnefs, as will appear from what follows. Since z (rr AB) is found rr \d X hyp. log. ~a by taking the fluxion thereof, we get k zz czdw^ I 4- wz aa ccdQ aa F 2ccQ au F 2c2 Q (becaufe Q zz w^ 1 F ^2)« Therefore j> (=: |
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