3HE. W. SPOTTISWOODE ON THE CONTACT OF CURVES. 
45 
Proceeding by the same principles, we shall have merely to replace □ by D in (15.), 
and there will result a system of expressions for the constants of the curve V having an 
m-pointic contact with a given curve U. 
The following method of reduction may sometimes be used with advantage. Let 
I, fj, . <p, X'> stand for the powers and products of the variables forming the various 
terms of V. Then the coefficients of the curve V having an m-pointic contact with U 
will be proportional to the determinants of which the following is a type ; 
But this 
=(-)^2D" 
. D’”-''v|/ 
= -2D'”-"|D 
D'”-";? D'"-% 
. D’”-"-v|. 
. D’”-"(p 
4 ' 
D2 D'”-";jD 
D'”-% . 
D— 
D'”-y D’"-^'^/ 
^ • 
X 4 ^ 
D"‘-% .. 
X ^ 
=(—)’"-" 2D'”-"| 2D'”-";? .. 2D(p D” 
D^ D"!.^ I 
X 4 ^ I 
In the case of the conic of 5-pointic contact this becomes 
( 21 .) 
- 2D V2Dy 2Dz"D" 
2Bzx 2D.ry 
zx xy 
I now proceed to the calculation of the conic of 5-pointic contact. This may be 
effected by any of the three methods indicated above, viz. (1) by means of the symbol □ , 
getting out the factors {ccx-\-^y-\-yzy' ; (2) by means of the symbols D; (3) by the 
reduced formula (21.). As most of the steps have been calculated by two methods, I 
subjoin some of the leading formulee and transformations which occur in the process. 
