An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

172

ORDERS OF CONTACT.

Orders of Hence it is that when two curves meet, and at the point
contact depend upon dy
the differ- of occurse have the same value of they are said to have
ential co- dx d 2
efficients., the ar e id th
efficientscontact of the first order, if moreover the same value -
dxZ
contact of the second order, and so on; and in general, if
dy dry dny
they have the same values of, d.. -, they are said
dx9 dxo2 dxr1
to have contact of the nth order.
onant of 243. If p (a) -f (a), '(a) - f'(a) *.. (a) -f (a) be
orderisac- each zero, and f"n+(a) -f"+l(a) not zero, then by (121),
companied \
with inter- p (x) -f(x-) has the same sign as
section,
whereas \n+1
cwheteof as X + (a) - f (a) (x - a)n+l
contact of
an odd
order is not. for all values of x sufficiently near a: now if the contact be
of an even order n is even, and n + 1 is odd, and therefore
~?(Ze) -f(x) changes its sign when x passes through the value
a; therefore, at one side of the point P, (p(x) is greater than
f(x), and at the other side less; i. e. the curves cross each
other at P, as in fig. 31. But if the contact be of an odd
order, n is odd, and n + 1 even, and therefore QQ' does not
change its sign when x passes through the value a; i. e. the
curves meet each other without crossing at the point P, as
in fig. 32. Hence contact of an even order is accompanied
with intersection, but contact of an odd order is not.
What has 244. It is evident that all we have just said is equally
been said
is true for true whether the co-ordinates be rectangular or oblique.
oblique and Moreover it is easy to see that if we refer to curves to any
polar coordinates new axes of co-ordinates, the degree of contact is not altered.
also.
The order It is clear also that what we have said applies equally to
of contact
is not dr d2r dnr
changedby polar co-ordinates; i.e. if -, have the same
a change of dO d dO
co-ordinates. values in both curves at the point of occurse, the contact will
be of the nth order: for then it is easy to see, putting
dy d'y dn'
= r cos, y = r sin, that will also have
the same values in both curvesd
the same values in both curves.