460 Mr Dixon, On the Order of Certain Systems of Conditions. 



conditions to be satisfied by the constants rj 1 , r) 2 . . . in order that 

 Q' = may have q contacts with S' = at points near the q nodes. 

 Suppose that in order to find the intersections of the two 

 curves we have eliminated y from Q' = 0, S' = 0. The result may 

 be written US' (x, Y r ) = 0, where Y 1} Y 2 ... are the different values 



r 



of y given in terms of x by Q' = 0. Move the origin to A 1} then 

 near A x x and y are small, and we have from Q' = a first approxi- 

 mation of the form 



y = ax + /3tj 1 . 



Hence when x is small one of the factors of the ^/-eliminant 

 becomes S' (x, ax + firji) approximately. The most important 

 terms in this, when x, ■q 1 , v 2 •••> e l5 e 2 ... are all small, are of the 

 form 



ax 2 + 2bxr] 1 + crji* +fe u 



and the x- discriminant of the ?/-eliminant will have 



a (fe 1 + en?) - &V 



as a factor in its most important terms. Putting this equal to 

 zero we have as the condition for a contact near A x 



Vi = ± A, Ve! + higher terms, 



X being a constant coefficient. The conditions for contacts near 

 the other nodes are of the same form, and there are q ambiguous 

 signs ; that is, there are 2? solutions of the contact problem which 

 merge into the single one 



Vl = V* = • • • = Vq - 0, 



when ei, e 9 ... all vanish. 



