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IN CENTRIPETAL FORCES.
173
V
icction /____!A will become ~-j rr AS ; which call a, and
let this value be fubftituted in thofe of AB and DE, and they
4ha hsz
will become —— and -r refpectively.
Hence if from the point Q, where the line of direction AC
cuts a femicircle defcribed upon AS, the lines SQ and QP be
drawn, the latter perpendicular to AB, the triangles ASQ and
AQP being fimilar, wc fliall have
r : s :: h (AS) : - rr AQ
r
r:s::-(AQ):~ rrPQ rr DE
r r-
r : c :: - (AQ) : ~ rr AP rr ^AB.
r i
PROPOSITION VI.
211. To determine the initio of the forces, whereby bodies,
tending to the centres of given circles, are made to revolve
in. the peripheries thereof.
Let ABH and ahh be any two propofed circles, whereof let AB and ab be
fimilar arcs; in which let the velocities
of the revolving bodies be refpectively as
V io v ; make DBK and dbh parallel to
the radii AC and ac, putting AC rr R, ac zz r, and the ratio
of the centripetal force in ABIT to that in abh as Ftof.
It is plain, becaufe the angles ABD and abd are equal,
that the velocities at B and b, in the directions BK and bk,
with which the bodies recede from the tangents AD and ad,
are to each other as the abfolute celerities V and v (art. 35).
But thofe velocities, being the effects of the centripetal forces
acting in correfponding, fimilar, directions during the times
of defcribing ABand ab, will therefore be as the forces themfelves when the times are equal ; but when unequal, as the
forces and times conjunctly. Therefore, the times being
• r » AB ab R r
univerfally as -prto—, or as —to— (becaufe the ares AB
V
v
1/
V
R
and ab arc fimilar), we have/as F x jr : f x
r
v
V ; tz.' ;
| 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 173. |
| Format | tiff |
| Identifier | 1138_201 |
| 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 CENTRIPETAL FORCES. 173 V icction /____!A will become ~-j rr AS ; which call a, and let this value be fubftituted in thofe of AB and DE, and they 4ha hsz will become —— and -r refpectively. Hence if from the point Q, where the line of direction AC cuts a femicircle defcribed upon AS, the lines SQ and QP be drawn, the latter perpendicular to AB, the triangles ASQ and AQP being fimilar, wc fliall have r : s :: h (AS) : - rr AQ r r:s::-(AQ):~ rrPQ rr DE r r- r : c :: - (AQ) : ~ rr AP rr ^AB. r i PROPOSITION VI. 211. To determine the initio of the forces, whereby bodies, tending to the centres of given circles, are made to revolve in. the peripheries thereof. Let ABH and ahh be any two propofed circles, whereof let AB and ab be fimilar arcs; in which let the velocities of the revolving bodies be refpectively as V io v ; make DBK and dbh parallel to the radii AC and ac, putting AC rr R, ac zz r, and the ratio of the centripetal force in ABIT to that in abh as Ftof. It is plain, becaufe the angles ABD and abd are equal, that the velocities at B and b, in the directions BK and bk, with which the bodies recede from the tangents AD and ad, are to each other as the abfolute celerities V and v (art. 35). But thofe velocities, being the effects of the centripetal forces acting in correfponding, fimilar, directions during the times of defcribing ABand ab, will therefore be as the forces themfelves when the times are equal ; but when unequal, as the forces and times conjunctly. Therefore, the times being • r » AB ab R r univerfally as -prto—, or as —to— (becaufe the ares AB V v 1/ V R and ab arc fimilar), we have/as F x jr : f x r v V ; tz.' ; |
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