VOL. XXVIII.] PHILOSOPHICAL TRANSACTIONS. 17 



c. And hence, completing the calculation, it will be i = 



the curve may coincide with the axis, and it will be i: = -—m r* With the 



'' ^/(bb — aaj 



centre c, and radius cd = b, describe the circular quadrant dpe, fig. 17, and 

 make ca = .r, and erect the perpendicular qp; then, the arc dp being y, it will 

 be w = — TTH . =z - z. Hence w = - z, and z = - w. And making; a? =: b 



^ ^/(hh — xx) a -^ a ' 6 -^ ° 



= CD, in which case also it is 3/ =: the quadrantal arc dpe, and z = ad = -i^L; 



^* D E CD 



then it will he-^h = a X — , and therefore a = l X - — . Let there be there- 



CD 2de 



fore CD : 2de :: the diameter of a circle : the circumference :: d: c; and it will 

 be «a = ll X — . Substituting then this value for aa, it will be - x l X — 

 the length of the pendulum isochronous to the string. Let therefore d be the 

 length whose periodical time is 1, so will - >/ be the periodic time of 



the string. For the periodic times of pendulums are as the square roots of 

 their lengths. 



Corol. 1. — The number of vibrations of the string, in the time of one vibra- 

 tion of the pendulum d, is -^ y/- X -• 



Corol. 1. — Because - ■/- is given, the periodic time of the string is as y^ — . 



And the weight p being given, the time is as |/nl. Also the strings being 

 formed of the same thread, in which case n is as l, then the time will be as l. 



An Account of some rare Plants, observed lately in several curious Gardens, 

 and particularly the Society of Apothecaries' Physic Garden at Chelsea. By 

 Mr. James Petiver, F. R. S. N° 337, art. 5, p. 33. 



An enumeration of 108 plants at that time esteemed rare, with short obser- 

 vations on each. 



Celestial Observations, made at the Royal Observatory, Greenwich. By Mr. 

 Flamsteed, Astronomer Royal. N°337, art. 6, p. 05. 

 These observations, made in the years 1711 and 1712, are on the sun and 

 moon, and the three superior planets Saturn, Jupiter, Mars; but being all 

 contained in the author^'s Historia Ccelestis, it would be of no use to reprint 

 them in this work. 



An Account of the Roman Legions. By Dr. W. Musgrave. N° 337, art. 7, p. 80. 



From this account we may observe, 1. That the number of legions under 

 the Roman consuls, at one and the same time, was greater than under the 



vol. VI. D 



