12 PHILOSOPHICAL TRANSACTIONS. [aNNO 1713. 



as in prob. 1. Let c be the centre of rotation in this plane, or at least its 

 projection made by a line demitted perpendicularly on the plane; and let a be 

 the point sought. Through c draw c^ at pleasure, in which take two points 

 z and ^, so that drawing zq and ^q, the angle czq may be obtuse, and the 

 angle cga acute; and in the points z and ^ let there be the particles p and tt. 

 Then draw zr and gr perpendicular to c^, which may be to each other as cz 

 to c^, by which will be represented the absolute velocities of the particles p 

 and TT. But certain parts of these, which are in the directions zq and ^q, are 

 taken away by the resistance of the point q. Draw cd and c^ perpendicular 

 to az and ag, then because of the equal angles zcd = rza, and ^cd = r^a, 

 the other parts of the velocities, in the directions perpendicular to az and a^, 

 will be as zd and ^d. Hence having the ratio of the distances qz and q^, 

 the forces of the particles p and tt, to move the space ab towards opposite 

 sides, will be as dz X za X jb and d^ X ^Ql X p. And by the conditions of 

 the problem the sums of these contrary forces ought to be equal. 



Because of the right angles at d and dj the points d and d are in the cir- 

 cumference of a circle described on the diameter ca. Let e be the centre 

 of this circle. Then drawing ez and Eg, meeting the circle in f, i and f, /, it 

 will be Dz X za = Fz X zi = ep'^ — ez'^ = eq^ — ez% and c?^ x ^Q = e^^ 

 — Ea^. Therefore the sum of all the eq^ X p — ez^ X p = the sum of all 

 the E^^ X TT — EQ^ X ^; and transposing the terms, the sum of all the eq^ X 

 (P+tt) = the sum of all the ez^ X j& + e^^ X tt; that is, if p be put as well 

 for the particle p within the circle, as for the particle tt without it, the sum 

 of all the EQ* X p will be equal to the sum of all the ez'^ X />. Draw zs per- 

 pendicular to CQ. Then will ez^ = cz^ + ec^ — qc X c^. Which value of ez^ 

 being substituted for it, and the equation ordered, there is at length found 

 the sum of all the ca X cs X p = the sum of all the cz* X p. Hence 



ca = ■sumofallthec z^x_y^ ^^^ ^^^ ^^^ ^f ^^^ ^^a ^ ^ jg ^j^^ quantity C itsdf, 

 ^ sum of all the c* X ;> '^ ^ ^ t-^v-n, 



in the calculation of the centre of oscillation : and if the centre of gravity 

 be G, and g^ be drawn perpendicular to cq, and the body itself be called a, 

 then will the sum of all the cs X p = eg X a. Hence is ca = 



Let o be the centre of oscillation; then by theor. 1, co = 



c 



eg X A ' CO X A 



Hence it is eg : cg :: co : cq. Therefore a perpendicular to co drawn through o, 

 will pass through the point q. q.e.i. 



^n Account of the Eruption of Mount Vesuvius, in 1707. By S. Faletta, 

 N° 337, art. 3, p. 22. Translated from the Latin. 



The eruptions of this mountain are so frequent and continual, that they are 



