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VII. — On the Motion of a Heavy Body along the Circumference of a Circle. By 



Edward Sang, Esq. 



(Read 20th March 1865.) 



In the year 1861 I laid before the Royal Society of Edinburgh a theorem con- 

 cerning the time of descent in a circular arc, by help of which that time can be 

 computed with great ease and rapidity. A concise statement of it is printed in 

 the fourth volume of the Society's Proceedings at page 419. 



The theorem in question was arrived at by the comparison of two formulae, 

 the one being the common series and the other an expression given in the "Edin- 

 burgh Philosophical Magazine" for November 1828, by a writer under the signa- 

 ture J. W. L. Each of these series is reached by a long train of transformations, 

 developments, and integrations, which require great familiarity with the most 

 advanced branches of the infinitesimal calculus ; yet the theorem which results 

 from their comparison has an aspect of extreme simplicity, and seems as if surely 

 it might be attained to by a much shorter and less rugged road. For that reason 

 I did not, at the time, give an account of the manner in which it was arrived at, 

 intending to seek out a better proof. On comparing it with what is known in 

 the theory of elliptic functions, its resemblance to the beautiful theorem of Halle 

 became obvious; but then the coefficients in Halle's formulae are necessarily 

 less than unit, whereas for this theorem they are required to be greater than unit. 



The search after the mutual relations of the two theorems has led me to the 

 discovery of a few simple propositions which involve only the very first principles 

 of the calculus, and the well-known law that the square of the velocity which 

 a heavy body acquires in descending along a curve is proportional to the vertical 

 distance, and to the intensity of gravitation jointly ; and which, yet, contain the 

 whole theory of motion in a circle whether that motion be oscillatory or con- 

 tinuous. I am thus enabled to present this hitherto intricate theory in a form 

 which renders it intelligible to junior students of mechanical science. 



By a well-known method of extension, the doctrine of the motion of a heavy 

 physical point along the circumference of a circle can be made to include that of 

 the rotation of any mass of matter on an axis not passing through its centre of 

 gravity, whether that axis be horizontal or be inclined ; hence, in the following 

 investigation, I may confine my attention to the motion of a physical point in the 

 circumference of a circle placed vertically. 



2. Let N be the nadir and Z the zenith point of a circle, along the circum- 

 ference of which a minute heavy body is free to move. If that body be projected 



VOL. XXIV. PART I. R 



