at an indefinitely - little equal Distance from, and parallel to MP; and fuppofe the indefinitely fhort Line IPPF=a. Solution. By the Property of an Ellipfe xx: b-x: PMq::b. p; pbx-px x confequently PMq= b In like manner will be found FGq = :x+a:x:b-x+a: xp, b pb x − p x x + 2pxa+pba-paa b The As TPM, TFH, TIL are fimilar; wherefore t t (TPq)" FHq pbx-px x = b (PMq) :: tt+2+a+aa (TF1) pbxttpxx t t + 2 p b xt a pb xa apxx à d pb. xt a 2 p ttb TI But FHq is FGq, and ILqIK9; that is pbxt t-pxx tt2pbxta 2pxxta + pb xaa — ttb рххаа pb x − p x x + 2 px a + b pa-paa b which Step, after due Reduction, gives 2 bx 2 x x + bx+tt PROP. IV. To draw a Tangent from the Afymptotes of an Hyperbola to its Curve. bola to be given; T 'tis required to find the Sub-tangent UT=t=? Note, UD being drawn || AQ, one of the Afymptotes, the Property of the Apollonian Hyperbola, in relation to its Afymptotes, is pp AUX UD, where p reprefents a determinate Quantity, and confequently may be expreffed by unity; fo that the fame Equation may become 1=AUX UD; that I is, 1x x UD, or x= UD=UD1. Whence all forts of Hyperbolas may be fet forth by this general Property AU— x= UD UD". Then Draw the Tangent TD, which produce to K, and imagine the Lines LKI and HGF, interfecting the Curve in Land H, the Tangent produc'd in K and G, and the Afymptote AT in I and F, to be drawn at an a, or indefinitely little equal Distance from and DU. Then Solution. UAx being (by Suppofition)=UD]",x m is there fore= UD; Alfo AI=x-a=IL" (by the Property of the Fig.); I therefore xam IL: And, for the like Reason, I x+a_m=FH. = The As TDU, TKI and TGF are fimilar; wherefore raifing each Part to the mth Power) x Fa therefore (by x I a ad= ; that is to fay (fince m хх ad is, as it is manifeftly, by the preceding + 30 I From the last, and last but two Steps, is N. B. In thefe Operations, be fure to take Care that your Multiplicator or Divifor be always o; otherwife the Majority will not hold, as in the foregoing Step: Thus, if h bek, and you multiply each by 1, if 1 beo, then 1h will be lk; but, if I beo, then 1h will be lk. CHAP. TO divide the given Quantity b, into two fuch Parts, that their Product may be a Maximum. Suppofex one Part, then bx : x:xx=6x b, and then b to the other Part of xx Maximum, Solution. increafed Suppofe the Part x to be diminish'd by the indefinitely small, or less than any affignable Quantity a, then the other Part of b will be b-x+a; And: x+a:x:b-x+a:=bxx x + b a + 2 xa-aa. But, becaufe bx-xx is (by Suppofition) a Maximum; therefore it is b x − x x + ba+ 2 x a¬aa: Whence o+ba+2xa-aa; and +2x+a+b; that is to fay, +2x+a+b and 2 x + a = b. : + 2 x − a ¬ +b. Now it appears by the laft, and the laft but two Steps, that let a be ever fo fmall a Quantity, provided it be more than Nothing, if it be fubtracted from 2 x, the Remainder will be lefs than b, but if it be added to 2 x, the Sum will be greater than; confequently, by our Lemma 1. 2x= = b; and x = b. PROP. II. To divide the given Quantity b into three fuch Parts, as, they being multiplied together, fhall produce a Maximum. Suppofe b-x to be one of the Parts required, then x will be the Sum of the other two Parts: But the greatest Product that can be made of any two Parts of x is, by the foregoing Propofition, x 2 x X : The Question propos'd is there 2 Fa; But the former being a Maximum, is therefore greater than the latter; that is + b x x 00 3 4 b x x − x3± 2 b xa + 3 x2 a+ba a − 3 x aa+a2; where 4: ± 2 b x a + 3 x x a + b a a — 3 x a a ‡a'. And, fore o Hence 3 x abad; And and 3 x x 0 - 2 b x • : 3 x x = 2 bx; confequent ly (from the laft, and last but two Steps) 3 x x = 2 b x, by our Lemma 1.; Wherefore of the Parts required= চে b; therefore one and each x = ) b − z b = ÷b. PROP III. m If n and m be known Quantities, and nm, and if xTM -x" Maximum: It is required to find the Value of x, it being affirmative. Suppofe x to be augmented by the indefinitely-little Quan decreased tity a; then x-x" will become x+am_x+a": xm. But the former being a Maximum, is therefore greater than the latter; that is (by Sir If. Newton's Theorem) xm - 201 |