254 



KANSAS UNIVERSITY SCIENCE BULLETIN. 



When b is near to a the coordinates of the line ab 



become 



ai 



a-i 



at + AOi ttj + AOj 



= P 



A Ui 

 A Ui 





AS AS 



where A^; indicates an increment of a,. 

 At the limit these homogeneous coordinates become : 



But these are the coordinates of a line through two 

 consecutive points of the curve ; that is, a tangent to 

 the curve at a. So that at a double point we have 

 an invariant lineal element. 



Similar considerations show that at a triple point we 

 have an invariant plane having only this point in com- 

 mon with the rest of the invariant figure. And at a 

 quadruple point we have an invariant space having only 

 this point in common with the remainder of the in- 

 variant configuration. 



What happens when the rank of this matrix is less 

 than four? 



In the first place, if the rank of the matrix is less 

 than four for some value of (;, (j, say, we know that p, 

 is at least a double root, for if the first minors of A (pj 

 become o, A'((j,) = o, which is the usual criterion for a 

 multiple root. Should the rank be lower than three, 

 A " (p j) = , and we have a root of at least multiplicity 

 three ; and so on. 



What are our invariant figures in these cases? For 

 the case of a matrix of rank three, the solution of three 

 of the equations (3) of section 1 gives us the point or 



