XLVI. On Clock Escapements. By Edmund Beckett Denison, Esq., M.A., 



of Trinity College, Cambridge. 



[Read November 27, 1848.] 



In the year 1827 the present Astronomer Royal wrote a paper in the Cambridge Phil. Trans., 

 Vol. III. p. 105, "On the Disturbances of Pendulums and the Theory of Escapements," in which he 

 investigated the effects produced on a free pendulum by connecting it with each of the three classes 

 of escapements ; and from the amount of the disturbance in each case, he inferred the relative merits 

 of the escapements. He added : " The Theory of Escapements is by no means complete, but I hope 

 it will be found that the principal points have been touched on, and that enough has been said to 

 enable any one else to pursue the subject as far as he may wish." 



I know of no work in which the subject has been pursued further ; and therefore I propose to 

 exhibit a few of the results which are to be obtained by following up Mr. Airy's calculations, and 

 which I arrived at in investigating the merits of an improved remontoir or gravity escapement, in- 

 vented and constructed by a friend of mine*, avoiding certain mechanical objections to which such 

 escapements have hitherto been liable; and it will be seen, from the following remarks, that they may 

 be made, by a particular arrangement of the parts, free from the mathematical objection which Mr. 

 Airy says renders them almost as bad as the common recoil escapement. Mr. Bloxam has had some 

 communication respecting his clock with the Astronomer Royal ; and 1 shall be glad if he is thereby 

 induced to complete his Theory of Escapements. In the mean time the following remarks may be 

 of some use. I shall take the mathematical results, though not the practical conclusions, of Mr. Airy's 

 paper for granted, as they are sufficient for my purpose ; and his method of obtaining them may be 

 seen either in the volume referred to, or in Pratt's Mechanics, into which the substance of his paper 

 has been copied. 



I shall presume that every one who is at all acquainted with clocks understands the construction 

 of the Dead Escapement, as it has superseded all others in clocks that are expected (as tile clock- 

 makers say) to perform correctly ; though it does not appear to be generally known from what the 

 accuracy of its performance really arises. I shall follow Mr. Airy in assuming the maintaining force 

 to be constant, although it is not quite so, since the inclination of the tooth of the escape-wheel to 

 the face of the pallet is greater at the end of the impulse than at the beginning, by nearly the angle 

 which the wheel moves through in one beat. Let /3 be the angle which the faces of the pallets make 

 with their dead or circular part ; then, since the tooth ought to be a tangent to the diad part, /3 will 

 also be the inclination of the tooth to the face of the pallet at the beginning of the impulse ; and we 

 shall assume it to remain the same throughout the impulse. 



Let Pg be the moving force of the clock-weight referred to the extremity of the oscnpc-wheers 

 teeth : p the length of tlie pallet measured from the axis of suspension of the pendulum : M the 

 mass of the pendulum, and / its length : 9 its angle with the vertical. Then the equation of 

 motion is 



say, (neglecting the moment of inertia of the whccK .iml puttin-; for sin as usual). 



• J. M, Bloxiiiii, Eb(i,, of LincnIn'M Iiiii, Il(irriiitcr-nt- hnw. 



