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[[left margin]] Remarks on Prob. I. p. 126. of [[underline]] Ludlam's [[/underline]] Astronomical Observations, made in St. John's College, Cambridge, in the years 1767 & 1768. where he computes the force of the swing wheel upon the pallets of a clock. See p. 300. [[/margin]]
It is well known that the swing  wheel of a clock [[insertion]] ^ would [[/insertion]] [[strikethrough]] acts upon [[/strikethrough]] [[insertion]] ^ urge [[/insertion]] the pallets with all that force or power, which it receives from the weight, by means of the other [[insertion]] ^ face of [[/insertion]] wheels and pinions [[insertion]] ^ if it acted perpendicular to the pallet --- [[/insertion]]; but  [[insertion]] ^ as the [[/insertion]] [[strikethrough]] by reason of [[/strikethrough]] the [[insertion]] ^ power [[/insertion]] [[strikethrough]] plane [[/strikethrough]] of the  [[insertion]] ^ acting [[/insertion]] tooth [[strikethrough]] being [[/strikethrough]] [[insertion]] ^ is exerted in a direction [[/insertion]] at right-angles to the plane of the wheel, [[strikethrough]] and [[/strikethrough]] [[insertion]] ^ which is inclined to [[strikethrough]] the plane [[/strikethrough]] [[/insertion]] the plane of the pallet's face, [[strikethrough]] and inclined to each other this action [[/strikethrough]] [[insertion]] ^ it [[/insertion]] will be exerted in an oblique direction on the pallet; [[strikethrough]] and [[/strikethrough]] [[insertion]] therefore will be [[/insertion]] part of it lost, and [[insertion]] ^ the remainder [[/insertion]] [[strikethrough]] part of it [[/strikethrough]] [[insertion]] ^ will be [[/insertion]] communicated to the pallet itself, [[strikethrough]] causing it [[insertion]] which shall cause its tension [[/insertion]] to turn in a[[/strikethrough]] [[insertion]] giving it a tention to move [[/insertion]] [[line]] [[strikethrough]] direction [[/strikethrough]] perpendicular[[insertion]] ^ ly [[/insertion]] to the plane of its face: 
[[strikethrough]] The resistence it makes [[/strikethrough]] but as it is not at liberty to move [[strikethrough]] to the ^ [[insertion]] against the tooth in [[/insertion]] [[strikethrough]] to the [[insertion]] ^ against the tooth in [[/insertion]] receiving this power from the tooth, causes a pressure upon the arbor [[/strikethrough]] in this direction, it makes a resistance against the tooth in receiving this force or power [[strikethrough]] of the Swing Wheel, and as it is not at liberty to move in a direction to [[/strikethrough]] and thereby occasion's a pressure both upon the arbor of the swing-wheel, and arbor [[strikethrough]] the perpendicular to the plane of its face, it causeth a pressure up also [[/strikethrough]] of the pallets themselves. [[strikethrough]] upon the arbor of the pallets. [[/strikethrough]] 
The remaining part of [[strikethrough]] the [[/strikethrough]]  [[insertion]] ^ [[strikethrough]] its [[/strikethrough]] this [[/insertion]] power, received from the tooth of the wheel, is exerted in [[insertion]] the [[/insertion]] [[strikethrough]] a [[/strikethrough]] direction of a tangent to the circle, which the point of action endeavors to describe round the arbor of the pallets. this arbor has therefore a kind of rotary motion, which is communicated to the rod of the pendulum by means of the crutch. Hence it necessary follows. 1[[superscript]] st [[/superscript]]. That if a power be impressed upon the plane of the pallet, equal, [[strikethrough]] to [[/strikethrough]] but in a contrary direction, to that which exerts itself in the aforesaid tangent; the whole movement will be sustained in equilibro, as proposed by M[[superscript]] r [[/superscript]]. Ludlam. 2. It is plain that whether the face of the pallet acts against the tooth, or the tooth is said to act against the face of the pallet, [[strikethrough]] to [[/strikethrough]]  [[insertion]] ^ that [[/insertion]] occasions the pressure upon the arbor of the swing wheel; yet this [[strikethrough]] force [[/strikethrough]] pressure will [[strikethrough]] will [[/strikethrough]] be exerted from the acting point [[strikethrough]] on [[/strikethrough]] upon the pallet or tooth, to the center of the wheel, and not from the center of the wheel to the pallet: and therefore does not conspire with, but opposes the action of the tooth upon the pallet, just the same as if it was exerted in a contrary direction upon the pallet itself: [[strikethrough]] for [[/strikethrough]] [[insertion]] ^ because [[/insertion]] it arises entirely from the resistence of the pallet alone. Wherefore that [[strikethrough]] part of the [[/strikethrough]] power which is communicated to the pallet, by the action of the tooth, is evidently divided into three others, [[underline]] viz. [[/underline]] one [[strikethrough]] at [[/strikethrough]] upon  the center [[strikethrough]] of the pa center [[/strikethrough]] of the swing wheel; [[strikethrough]] and [[/strikethrough]] [[insertion]] ^ another upon the center of the [[/insertion]] arbor of the pallet, [[strikethrough]] and a third [[/strikethrough]] in a direction from the acting point to those centers; and a third, [[strikethrough]] which is [[/strikethrough]] communicated to the pendulum by the crutch, in the direction of the aforesaid tangent. 3. When a power is impressed equal, but in a contrary direction, to this last mentioned power, which tends to move the pendulum, and the whole movement [[insertion]] ^ is [[/insertion]] in equilibro, as above expressed; it is manifest, that the power impressed upon the tooth of the swing-wheel, by the weight, counteracts, and sustains the other three; viz. [[insertion]] ^ one upon each arbor, [[/insertion]] [[strikethrough]] two upon the two arbors, [[/strikethrough]] and another supposed to be impressed upon the pallet; [[strikethrough]] besides [[/strikethrough]] [[insertion]] ^ over and above [[/insertion]] what is lost by reason of the oblique direction. 4. [[insertion]] Hence it [[/insertion]] [[strikethrough]] It [[/strikethrough]] cannot [[strikethrough]] hence [[/strikethrough]] be denied, that this power, impressed upon the tooth of the swing wheel, by the weight, is greater than the whole sum of the other three, which counteract, and keep it in equilibro. - These [[strikethrough]] are [[/strikethrough]] plain, and evident matters of fact, not involved with any mathematical theory, will greatly contribute to our assistance in the application of mathematical principles, for [[strikethrough]] the [[/strikethrough]] comput[[insertion]] ^ ing [[/insertion]] [[strikethrough]] ation of [[/strikethrough]] their [[insertion]] ^ respective [[/insertion]] effects: 
For I believe they are the standard of truth, and will bear the test of comparing and measuring what [[insertion]] ^ =ever [[/insertion]] has been [[insertion]] ^ truly [[/insertion]] said on the subject; [[strikethrough]] where [[/insertion]] in order to obtain a right judgment. Those

Transcription Notes:
@siobhanleachman - reviewed some of this but going to have to come back to it.