53.2 Mr. S. Roberts on the Order of the Conditions that 



curves (Crelle, 1866, part 4). I have thought it worth while to 

 give a proof apart from geometrical considerations, and to apply 

 the results to some cases in which the coefficients of the equa- 

 tion are supposed to be functions of secondary variables. Let 

 it be supposed that the coefficients of x m , x m ~\ x m ~ 2 , &c. in the 

 given equation are of the orders X, \ + a, \-\-2u, &c. in the 

 secondary variables. 



According to a method elsewhere indicated*, I suppose the 

 equation to be broken up into factors of the first degree, the co- 

 efficients being of the orders -> — \- « respectively. The order 

 ° m m 



of the conditions for the stated. set of multiple roots must be of 

 the form 



m . m— 1 . . . m —p -f 1 ^ m.m — \ . . . m —p + 2 ^ "\ 



~~1.2...j» -■**+- i.2..j>-l p " V (e) 

 + &c. +mK l , J 



where p is written for l^n + 1, K ? , is the order of the conditions 

 that p factors of the given equation may constitute a set of t 

 multiple roots of the orders n v n 2 , . . . n t respectively without the 

 necessary evanescence of any factor; K p _ x is the like order rela- 

 tive to p — 1 factors of the given equation, the evanescence of a 

 factor being necessarily implied, and so on. 



It is important to remark that Kp_i, Kj,_ 2 , . . . K, are all 

 affected with the multiplier X, since in each case the identical 

 evanescence of a factor or factors is implied. 



Now Kp is given by 



\m / \m ) \m J \m J 



in+t 



since we have only to apply the general theorem* for the 

 order of the conditions that a given set of linear equations may 

 coexist to the sets of n Y -\-\ factors, w 2 + l factors, &c, nt+1 

 factors into which ^n-\-t factors may be divided. But there 

 are 1 . 2 . . . Sw + 1 such divisions, which gives the first portion 

 of the expression for K^,; and the rule or theorem referred to gives 

 the remaining factors of the expression. 



This being so, it follows that if \=0, a=l, the order of 

 which we are treating is 



(m, (m — 1) . (m — 2) . . . m — Xn — /-f-l); 

 * Salmon's ' Modern Algebra/ 2nd edit. p. 281. 



