TANG 



Itiuds 9^ elUpfcs, (putting a aad c fur llie two priii- 

 dpal ^ametws) is a-x^ x x" = ^ x jr"' * ", vre (hall 



l»ve -mix a — x^" - \ x. x" + n x x" ' xa — x|' 





•^ m + h X y"+'- 'y; and, therefore, 



X m + n X 31"' 



UL — 



m + n X a — « ) " X x*^ 



_ m x" X a - .v'>'" - ' + n.x"- ' X a- *' 



. m-\-nXa — xXx m-r-tiy.ax — x'' 



,- + ■. = a _ xl"- X «") = ; = - 



J -mx+nxa — x na — n+mxx 



=r the fubtangent required. 



VI. The equation defining the hyperbola is c' x 

 f X + x' = a'' y', a and c being ufed to denote the two 

 principal diameters ; whence we have, c^ x ax + z k x 



X ^^ y V X 



= 2 «' »i ; confequently — = ^ — ■ and - — =t 



y c' X ^a +x y 



a-y- 



r X ax + X _ 



ax -{- X- 



_ = the fubtangent ; 



e' X ^a + X c'' X ^a + X \a + x 



whence the diftance of the point of interfeftion of the 

 tangent and axis from the vertex, which is equal to the 

 difference of the fub-tangent and abfcifs, may be found ; 



ax + x^ I „ ^ 



for ==^ - * = T^ 



■ ; and, therefore, that point 



ExNT. 



The preceding examples relate to curves, whofe ordinates 

 are parallel to each other. We fhall now briefly illuftratc 

 the method of drawing tangents to curves of the fpiral kind, 

 all whofe ordlnales iffue from a point : fuch as the fpiral BAG 

 ( Plate XV. Anal. Jig. I . ) whofe ordinates, C B, C A , C G, are 

 referred to the point C, called the centre of tlie fpiral. Let 

 SAN be a tangent to the fpiral at any point A, and let 

 C T be perpendicular to it, and let the arc C B A (confi- 

 dered as variable by the motion of A towards G) be de- 

 noted by 2, and the ordinate CA by j'. Then z:y :: 



AC (>) : AT =-''^. Hence, if upon C A, as a dia- 

 meter, a femicircle be defcribed, and in it, from A, a right 

 line equal to -r- be infcribed, that right line will be a tan- 

 gent to the fpiral at the point A. 



VIII. Let the nature of the curve C B A be fuch, that 

 the arc C B A may be, always, to its correfponding ordi- 

 nate C A in a conftant ratio, w'z. as a to i : then, becaufe 



X : y :: a : t, •we have z = -y-, and z ■ 



; and, confe- 



being given, the tangent may be eafily drawn. 



The manner of drawing tangents to all forts of hyper- 

 bolas umverfally, will be the fame as in the eUipfes, the 

 equations of the two kinds of curves differing in nothing 

 but their figns. 



After the manner above explained, the fubtangent, in 

 curves whofe abfciffes are right lines, may be determined ; 

 but if the abfcifs, or line terminating the ordinate, on the 

 lower part, be another curve, then the tangent may be 

 drawn as in the following example. 



VII. Let the curve B R F {Plate XIV. Anal. fg. 22.) 

 be a cycloid ; whofe abfcifs is here fuppofed to be the femi- 

 circle B P A, to which let the tangent P T be drawn, as 

 above. Moreover, let r R H be a tangent to the cycloid, 

 at the correfponding point R, and let G R e be parallel to 

 T P ■« ; putting the arc, or abfcifs, B P = z, its ordi- 

 nate P R ^ J-, A F = *, and B P A = ^ ; then, by the 

 property of the cycloid, we ihall have c (BPA) : b 



(AF :: 2 (BP) :;, (PR) ) ; therefore j, = ^^ andi = 



hi. 



— = re. But by fimilar triangles, r* (j) : R« (= P ^ 



= i) t: P R (jy) : P H = ^ = ^ (becaufe J, = i?) ; con. 



fequently, if in the right line P T, there be taken P H 

 equal to the arc P B, we (hall have a point H, through 

 i^lMct U^ tangent of the cycloid muft pafe. 



quently, A T {^-P\ = ^ = — x A C : therefore A C 



Snd A T being in a conftant ratio, the angle CAT muft 

 alfo be invariable ; which is a known property of the loga- 

 rithmic fpiral. 



IX. Let BAA {Jig. 2.) be the fpiral of Archi- 

 medes ; whofe nature is fuch, that the part E A of the ge- 

 nerating ordinate, intercepted by the fpiral, and a circle, 

 BED, defcribed about the fame centre C, is always in a 

 conftant ratio to the correfponding arc B E of that circle. 

 Suppofe A n perpendicular to AC; B C = c, C A = ^, 

 and the given ratio of A E to B E, that of ^ to c ; then 



I : c :: y — c {KY.") : -^—j — = B E ; whofe fluxion is = 



cy 



If the right line C E A a be fuppofed to revolve about 



the centre C, the angular celerity of the generating point A, 

 in the perpendicular direftion A n, will be to that of E, 

 as A C to E C ; and as the latter of thefe celerities is ex- 



preffed by -j-, the former will be exprelTed by -— x - 



*' 



jy 



or ■'-J ; which is to j, the celerity of A in the direftion 

 A a, as ~j- to unit, or as y to I. Confequently, C T and 

 AT are in the fame ratio, and AC : CT :: -^ yy + bb 



■.y; and AC : AT :: Vyy + i>b:b; whence C T and 



A T are given, equal to — — ^ , and 



'^ yy + ib "^ yy + ^^ 



fpeftively ; from either of which exprefllons the tangent 

 A T may be drawn ; aad, in the fame manner, may the 

 pofition of the tangent of any other fpiral be determined. 

 Simpfon's Flux. vol. i. feft. 3. 



As to the method of inveftigating tangents by fluxions, 

 fee Macl. Flux, book i. c. 7. where it is demonilrated in- 

 dependently of infinitefiraals. 



To 



