20 AN ACCOUNT OF THE EARLY MATHEMATICAL AND 



what is the locus described by the weight of the pendulum ? 

 The weights of the vessel and pendulum being given, and the 

 distance of their centres of gravity." 



Answered hy Mr, John Dalton, " Let p and R be the 

 centres of gravity of the vessel and pendulum when horizontal ; 

 W and IV their weights ; then from p towards R, take /jN : 

 NR :: i^ : W, and N will be the common centre of gravity 

 of the vessel and pendulum. 



The force of the pendulum "^h- 4 



upon the centre P at any instant 

 may be resolved into two others, 

 the one perpendicular and the 

 other parallel to the horizon ; 

 the first of which only tending 

 to increase the friction of the 

 vessel upon the plane, by hypo- 

 thesis, has no effect, and the other communicates the horizon- 

 tal velocity to the vessel. Moreover, as the action between 

 the vessel and pendulum is reciprocal, their common centre 

 of gravity cannot be made to deviate from the vertical line 

 NM; consequently the distance of their respective centres 

 P and G from that line will be always the same ; the pen- 

 dulum will coincide with NM after half one vibration, and 

 the whole space described by the vessel during one vibration 

 will be = 2j9N. Now let /?R = PG = NM = a ; and then 

 NR = a^ = TG = 6; also QG = x, and NQ = y. 

 Then PG = a : PQ = (a^—rr^)* :: TG = 6 : NQ = z/; a 

 known property of the ellipse, which is consequently the 

 locus required." 



Question 663. By Mr, John Dalton, " Two indefinite 

 right lines form a right angle at C: — from a given point A in 

 one of them draw lines to meet the other in D, on which set 

 off AM ; CD :: 1 : n : — required the quadrature of the curve, 

 which is the locus of the point M ?" 



Answer hy Mr. Gough. " The curve is a line of the 



