"a: 
_ 
Direct 
- Method. 
FLUXIONS. 
bola, and let F be the focus, and FE a perpendicular to 
the PE. Draw a straight line from E to A, the 
vertex of the axis; then AE is a tangent at the’ 3 
pee me aan Sect. IV. Prop. 14. cor. 1,) and there- 
ore FAE is a right 
FE =p, and the angle AFE = u, and observing that 
AF = } parameter = }. a, we have for the equation of 
_ the curve ~ . 
a= 
P= Teo. 
And taking the fluxions, by art. 30. and art. 26, 
ing that p is a function of u, 
dp__asin.u dip _ (costu+2sin.tu)a 
du” 4costu due ~~ 4cosiu : 
Therefore, by formula (A”), art. 88. 
d 2(cos.? u+-sin.? u)a 
19-4 T= 4 cos.3 u 
observ- 
@ . 
2cos3 u’” 
Ex. 2. Let the curve be any one of the conic sections. 
Ti Ae cogin. Ade cove tostes be taken stone srire- 
pri ys the equation 
Hence we find, Sala canoe 
_dy= fneenis, 
fay+ (m+4+2n2)? i dx* 
4y* A 
fa( me 4 nat) +(m +2n2)} dst 
= ¥ 7 > 
Qnydx2—(m+2nx)dxdy 
_ : 2y" a 
axis, their nature may be expres- 
d at4dy* = 
dy= 
4 {any — (m 4 2n2) } dt 
4¥ 
Hence, from formula (A), we get 
3 
{ast (m-p2enay} 
ca ny—Z (M4 2nz 
and substituting for y? its value, ; 
F a¢mix-t-ns*) +-(m-42n2)*} 
r= . 
: 2m* 
By giving to m and n the values that belong to the 
different curves, (Conic Sections, Sect. VIII.) this for- 
mula will give the radius of curvature in each case. 
90. It appears from what has been shewn in this section, 
that the contacts of curve lines may be accord- 
ing to different orders. The of contact, in which 
the ordinates, and also their fluxions, are equal, 
(art. 81.) may be called a contact of the first order ; 
and when in addition to these, the second fluxions are 
equal, (art. 82;) itis a contact of the second order ; and 
so 
There are curves which, with a given curve, admit 
only of contacts of a certain order. A straight line, for 
instance, is only capable of a contact of the first order. 
A circle may have a contact of the first, and also of the 
second order, but none higher ; and a curve, whose 
equation is y=a + br-+c z*+4d2°, is also capable of a con- 
led triangle. Hence, putting ' 
angl . Lm second requires three; and so of the higher orders. 
429 
tact of the third order ; and so on. ‘The degree of con- 
tact of which a curve is capable, depends upon the num« 
ber of constant tities to be determined, These may 
be called the elements of contact. A contact of the first 
order requires two constant quantities ; a contact of the 
In an analytical point of view, the contact of a straight 
line, or of a crcle eth a curve, is not more interesting 
than the contact of any other curve, unless on account 
of these curves being more elementary. The circle of 
curvature is, however, interesting, because of the sim- 
le geometrical expression it gives for the measure of a 
leflecting force. (Principia, lib. i. prop. vi. See also. 
Physical Astronomy, 
chap. i). 
91. The first formula which we have given for the 
radius of curvature, (art. 86,) has been investigated 
upon the hypothesis, the curve is concave towards 
the axis. In this case, 4 is a negative quantity ; 
and hence the sign of the expression for r is negative. 
If the curve had convex towards the axis, pe the 
sign of 7“, and of the expression for r, would have 
been positive. Upon the first h esis, 7 comes out 
a positive quantity in the applications of the formula, 
as in the examples we have given ; but when the curve 
is convex towards the axis, it has a negative value. 
Of the Evolutes of Curves.. 
92. Let LHM be a curve of any kind, (Fig. 24.) and 
let us suppose that a thread, fastened to the curve at 
some point beyond M, is. drawn tight, and applied 
upon it, so as to have the position CLOM ; that thismay 
be done, the curve may be conceived to be the common 
section of a plane, and some solid rising a little above 
it, round which the thread is wound. Suppose now, 
that while the thread is tight, it’ is ually un- 
from the curve. hile the portion between L 
and H is unwinding, its extremity, P will describeu 
the plane some line CP, and the process of unwinding 
being continued, a curve CPD will be generated, the 
nature of which will depend on the mode of its genera- 
tion, and the ies of the other curve L 
The curve along which the thread is wrapped, is eal- 
led the Evolute of the curve, generated by the extremity 
of the thread ; and, on the other hand, the latter curve 
is called the Involute of the former. Our present object 
is to shew, how the evolute of any proposed’ D: 
generated 
may be found. 
may immediately draw these three 
93, From the manner in which a curve is 
from its evolute, we 
conclusions : 
1. The to of the thread PH, which is disenga- 
ged from. the evolute, is a tangent to it at H. 
2. The straight line PH, is equal to the are CH of 
the evolute. 
3. Any tangent to the evolute, is a normal to the 
curve. In fact, any point H of the evoltte may be con- 
sidered as a momen centre; and the line HP ‘as 
the radius of a circle which the point P is describing, 
when the point of contact of the tangent’and curve is 
at H. It is from this last property, that we propose to 
deduce a solution of our prow ? 
Let AB be a common axis to the two curves, and let 
the normal PH produced meet the axis in N. Let 
P’H'N’ be another position of the normal, mecting the 
Direct 
Method. 
Fig. 2h. 
