16 AN ACCOUNT OF THE EARLY MATHEMATICAL AND 



Fig. 



v\ and t z= the time in seconds. Then c : 1 :: r ; ^ = t\ 



c 



per question. .*. t = (^)«. Again 1 : s :: t : st. .*. AB = 

 s 0)1 But BC2 = AB2 + zj2; and BC : AB :: PC : PM 



=: X = bs {~)i -^ {^^(O" + ^^}^5 which reduced by putting 



P = 2ir^ ^"d ^ = 1 ^ i • I x^ f I becomes y = v — 



(^ — x"y; the equation of the curve. 



If w = 1, then V vanishes, and x is constant, being = bs 

 -T- {b^ + c^y ; in vi'hich case the locus is a right line parallel 

 to AC. If n be greater than unity, p is affirmative, and x^ is 

 in the denominator of v or ^; consequently, when x = o, y = 

 00, and AC is an asymptote to the curve. If w be a proper 

 fraction, or p negative, the numerator of p becomes the deno- 

 minator; and when x •= b ■=■ AV, y = Qc, consequently a 

 line drawn parallel to AC through V, will in that case be an 

 asymptote. When n is greater than unity, ii x =: b, then y 

 = o, or the curve meets the line AB in the point V. When 

 n is less than unity, if a? = o, y = — b, or the curve meets 

 the line CA produced to R till AR = b; — and in all cases 

 the curve will cross the line AB, or it will somewhere be v = 

 MC = (6^ — x^y: — for, firstly, if jo be negative, then when 

 X = o, the equation gives x = b, but as x increases the first 

 side of the equation increases, and the last decreases, whence 



