CONTACT OF CURVES. 337 



Hence, since the curves have contact of the w th order, 



Now suppose y % (x) to be the equation to a third curve 

 which has contact of the m th order with the first curve at the 

 point x = a ; then if y s = % (a + h), we have 



If m be less than n we can give such a value to h as will 

 render y l y z less than y t y a for that value of h and all 

 numerically inferior values both positive and negative. Hence 

 in the vicinity of the common point the second curve is nearer 

 to the first than the third is. 



In the above expressions 6 denotes merely a proper fraction, 

 and it is not necessarily the same proper fraction in the 



different cases. 



318. The expression for y l y, i in Art. 317, when h is 

 sufficiently diminished, has the same sign as 



and therefore changes sign with h if n be even; therefore 

 if two curves have contact of an even order they cross each 

 other at the common point. If two curves have contact of 

 an odd order they do not cross each other at the common 

 point. 



319. In order that a curve may have contact of the 

 n th order with a given curve, it appears from Art. 316 that 

 n+l equations must be satisfied. Hence, if the equation 

 to a species of curves contain n + l constants, we may, by 

 giving suitable values to those constants, find the par- 

 ticular curve of the species that has contact of the n th order 

 with a given curve at a given point. For example, the 

 equation to a straight line is of the form y = mx + c ; since 

 there are two constants, m and c, we may, by properly de- 

 termining them, find the straight line which has contact of 

 the first order with a given curve at a given point. If the 



T. D. C. Z 



