[ « ] 
III. On the Contact of Curves. By William Spottiswoode, M.A., F.R.S. 
Eeceived October 15, — Eead November 21, 1861. 
Let U=0 be the equation to the curve with which the curve V=0 has an “ m-pointic 
contact;” in other words, let the cuiwes U and V have m consecutive points in common. 
The degree of contact of which V is capable is equal to the number of independent con- 
stants contained in its equation ; i. e. to the number of terms in its complete expression, 
less one. Thus if n be the degree of V, the degree of contact 'will be j 
If the curv es U and V have only a single point in common, then the only conditions 
are 
U = 0, V=0 • (I.) 
If they have two consecutive points in common, then beside (I.) we have also 
BU j , bU , . bU ^ „ 
av , , av , . av 
which, as is well known, lead to the conditions 
( 2 .) 
^ . w 
dx ■ d\j 
^ . -bff 
' dx ' dy 
dz 
dz ■ 
(3.) 
and if V be linear, =Ix-\-niy-\-nz, (3.) will suffice to determine the ratios l:m:n, and 
fix the position of the tangent Y. The conditions (3.) may be expressed by a single 
equation thus : if «, j3, y be any arbitrary quantities, then the equation 
= 0, (4.) 
a 
/3 
r 
W 
dx 
dy 
dz 
BY 
dx 
dy 
dz 
considered as identical in a, (3, y, may be regarded as an expression for the required 
conditions. 
The developed form of (4.) is 
dx 
bU _bu\ W / BU BU\ BY , /A 
dy ^ dz ) dx \ dz dx J dy y 
^ dy J dz ’ ’ 
( 5 .) 
MDCCCLXII. 
G 
