422 
Diveet gent were on the same side of the ordinate: as in the 
Fig. 12. 
Pig. 13. 
pomeavent Gauieseee we Se uan- 
tity, we infer that and the eatin es 
ies sides off the ordinate; So that; Without 
t 
positian, TQ="—"", from which it appears, that the 
subtangent is independent of the conj axis. In 
the hyperbola, the subtangent may be found by exact- 
ly the same process. 
Ex. 4 apogee (Fig. 12.) of which 
AB is the axis, and AHB ing circle. Let 
the origin of the co-ordinates be A, the vertex: and let us 
suppose that the radius of the circle is unity. 
Put the are AH =v ; then, by the nature of the curve, 
(see Ericyciom,) «= 1— cos.v, y=v + sin. v ; therefore 
dz=sin. vudv, dy=dv-+cos.vdv, therefore, 
dz _ ysin.o a, Y L008. 9) _ 
ay it awe inn ~=T2 
is, POX RO TQ; therefore, QH : QP:: QA: 
QT. Seog or recreate rethld are 9 
F of the generating circle, as was 
ved in the article Erircycuorp. en 
Ex. 5. In the logarithmic curve, of which the equa- 
tion is y = a*, we have dy=a"l.(a)dx=y 1.(a) da, 
de 1 
« 
dy? Lay 2 em. 
In this case, the subtangent is a constant quantity. 
69. When a curve 2% 
a polar equa- 
tion, we may first find the equation of the rectangular 
i as i ined m Curve Lines, art. 21, 
and then 
venient to 
is 
y the formule ; but it will be more con- 
e formule suited to that particular mode 
of expressing the nature of the curve. 
of the 
Let the centre 
ing radius AP=r, and the variable angle PAB=v, 
w it makes with the axis AB. We may now res 
gard the angle v as the i t_ variable qnantity, 
andAQ=2,QP=y,and AP=ras functionsof that quan- 
tity. Put ¢ for the angle PTQ ; then because the angle 
=v— tan. TPA = _“4n..v— tan. ¢ 
TEA oi -h neers Ae pinta ee 
(Anrrumeric or Sings, art.26.) But tan. v= ae = 
4, and we have found (art. 67.) that tan, i= oh 
Pa . ey 
therefore, 
44 
tan. TPA = 242 94s —sdy. 
dy” xdatydy™ 
14 ue uv y 
Now, by trigonometry, 
2=r cos. v, y=r sin, v; 
the fluxions, (art. 29. and art. 26. 
dzx=cos.vudr—r sin. vdv, 
dy=sin. vdr+rcos.vdv; 
and hence again, 
ydz=r cos. v sin. vd rr sin.*v dv, 
ady=r cos. vsin. vd r+7* cos.*vdv, 
2dz=rcos.tvdr—r*cos. v sin. v dv, 
yd yar sint odr +r*cos.v sin. vd. - 
therefore, 
Rule D.) 
FLUXIONS. 
Therefore, y ” sevid Sate ridu, 
adr =rar; 
oud koche’.  Paeeeh 
tan. TPA = — = 
var 
Through A, draw AT’ perpendicular to the ras 
dius AP meeting the in T’; and the 
point of contact P, draw PN perpendicular to the tan- 
gent, meeting T’A in N; We may [? 
as the sublangent and AN as the subnormal. be- 
cause, in the similar right angled triangles TAP, PAN, 
rad. : tan. TPA(= 5 ::PA(=r): AT’:: NA: PA, 
therefore. subtangent AT =—S' 5, (7) ' 
subnormal AN= = (aye ores 
v 
Ex. Let the curve be the spi 
70. In some curves, the distance between the ori- 
gin of the co-ordinates and the point in which the tan- 
pt dpe cn bseghprawhgere he | 
issa x is infinite, that distance becomes P 
SN Te aan. aur eel 
axis at a finite distance from the origin: It is then an 
Gi, % (Fig. 8) the abscissa AQ =< be subtracted, Fig. & 
the remainder eyo is the general expression for 
TA, the distance of the intersection of the tangent and 
axis from the origin’of the co-ordinates. If when < is 
infinite this expression is finite, we may conclude ‘that 
the curve has asymptotes, but if it be infinite, then the 
curve has no asymptote. 
Ex. 1. The equation of the hyperbola(Conrc Sections,. 
Sect. VIII.) is a? y? = 4? x*—a? b?, the origin of the co- 
ordinates being at the centre; in this case, atydy= 
xdx, and 
dz _ ay? _x—a? a 
y= prs : 
totes, which through A 
Ex. 2. The equation of the parabola is y*=a x, hence 
a2 ymax; when z is infinite, this quantity becomes 
infinite ; therefore the curve has no 
The method of tangents i i 
tigues, Lagrange; Traite du Calcul Di i i, 
Lacroix; A Treatise of Fluxions, arma Analyse 
OE 
