34 Mr. Squire on the Semi- arcs of Vibration of a Clock 



pal', ov nop is bisected by the line ol. In the same way it 

 may be shown that o?ip and 7ip o are bisected by vt n, kp ; 

 and hence tlie intersection d of these lines is the centre of the 

 circle inscribed in n op. 



Prop. XIII. (fig. 2.) 



The centre H of the circle circumscribing any triangle 

 ABC is equidistant from the centre e of the circle OPZ in- 

 scribed in the triangle, and the centre d of the circle FGE 

 passing through the three centres of the circle of external 

 contact. 



For d being the centre of the circle EFG, and e the inter- 

 section of the perpendiculars from its angles upon its opposite 

 sides, it follows from Pr. IV. and its corollaries, that if ed \s 

 bisected in H, the point H is the centre of the circle ABC. 

 Whence the proposition is obvious. 



Cor.— It is thence also plain that if ABC be any triangle 

 whatever inscribed in a given circle, the circle through the 

 three centres of the external contact is of an invariable mag- 

 nitude, viz. having its diameter double of that of the circum- 

 scribing circle. 



With these remarkable properties of the triangles ABC, 

 EFG, and their respective reciprocals, nop,kl m, I shall close 

 the present paper. I intend, however, so soon as I can com- 

 mand sufficient time to arrange my materials, to select from 

 the memoranda which have been accumulating upon my hands 

 for some years, two or three other series of propositions, which 

 I flatter myself will not be altogether uninteresting to geo- 

 meters. 



Bath, Jan. 15, 1827. 



VI. On the Semi-arcs of Vibration of a Clock with a Dead Beat 

 Scapement and DealPendidum. By Thomas Squire, jEsg-.* 



To the Editors of the Philosophical Magazine and Annals of 

 Philosophy. 

 Gentlemen, 

 TN a paper dated Dec. 15, 1824, and which appeared in 

 *■ vol. Ixv. page 38. of the Philosophical Magazine, I have 

 made mention of having registered several hundred observa- 

 tions of the semi-arcs of vibration of a clock with a dead beat 

 scapement and deal pendulum. Since that time I have con- 

 siderably extended this series ; and from which it appears 

 that these arcs are at times so unaccountably variable, that 

 for this reason I am induced to trouble you with some re- 



* Communicated by the xVuthor. 



marks 



