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XVI. — Additional Note on the Motion of a Heavy Body along the Circum- 

 ference of a Circle. By Edward Sang, Esq., F.R.S.E. 



(Read 19th December 1870,) 



In the twenty-fourth volume of the Society's Transactions, a very convenient 

 formula is given for computing the time of oscillation in a circular arc ; and the 

 investigation of that formula is conducted by an appeal to the actual pheno- 

 mena. It is defective in so far that it contemplates chiefly the time of oscilla- 

 tion over the whole arc, and does not enable us conveniently to compute the 

 time in which a part of that arc is described. 



The object of the present note is to supply that defect, and to present the 

 whole subject in a new aspect remarkable alike for its generality and for its 

 simplicity. 



Referring to the first figure given in the paper cited, let N be the nadir and 

 Z the zenith point of a circle placed upright ; and let us suppose a heavy 

 physical point to be projected from N along the circumference with a known 

 initial velocity, the object of our inquiry is to ascertain the law of its motion, 

 and to compute the time in which it describes a given arc. 



If the initial velocity be due to a fall through the height NA greater than 

 the diameter, the body will reach the zenith point Z with a velocity due to a 

 descent through ZA, and will continue its motion along the other semicircum- 

 ference, reaching N with the same velocity as at first ; thus it will circulate 

 along the whole circumference with a variable speed. 



But if the initial velocity be that due to a fall through NB less than the 

 diameter, the body launched in the direction N/3 will gradually lose speed until 

 it come to rest at the point F on a level with B. Thereafter it will descend 

 along FN, pass to the other side, and again return to N, repeating over and 

 over again its oscillation. 



In the original paper a connection was established between cases of con- 

 tinuous and reciprocating motion. This connec- 

 tion may be more neatly traced by the following 

 scheme : — 



Fig. 2. 



Let a trigon have two sides AC, CB of 

 definite length jointed together at C, and let 

 the angle ACB gradually change. Beginning 

 with AC, CB in a straight line, the angles 

 at A and C are each zero. As CAB in- 

 creases, ABC also increases, until, if CA be the shorter leg, CAB becomes 



VOL. XXVI. PART II. 6 A 



