61 6 Mr DE MORGAN, ON THE SINGULAR POINTS OF CURVES, 



To fix our thoughts, let us suppose lower degree is in question. Let m + signify 

 'w» or more': thus the dimension of the first term is n +, which is n unless vu = 0, 

 and is then p unless -wu also = 0, &c. And first let /uu = have no other roots equal 

 to the one selected for trial; that is, let fi'u have value. Remark that we must exclude 

 negative and zero dimensions of U as irrelevant; preceding processes having obtained w 

 only as derived from U = when x = 0. Using the general theorem, we examine 



(n +) - m (n +) - {m +) (« +) - (m +) 

 1-0 2-0 3-0 



where n + is of one meaning throughout. The greatest of these must be the first ; there 

 is then one root of the positive dimension (n +) - m which, if fractional, is of the form 

 k + kt, so that its denominator is that of r or a factor of it. Beginning with m and 1 , 

 we find in the next succession of fractions nothing but zeros or negatives, hence the next 

 form of U (and all which follow) must be rejected. The value of r / is then (w+)-m; 

 u is determined from 



(the first having value of vu, ts-u, &c.) + \lu . m / = 0. 



And in the subsequent processes, that is, in writing cc r -u for U t , and then m / + U ) for w /( &c. 

 it will easily appear that because fxu has value, and the first term of the first term vanishes, 

 the dimension of the second term is always, throughout the processes, lower than that of the 

 first, and at least as low as any one which follows. The results are then univocal throughout, 

 if we remember that a? r and its integer powers are taken univocally, and we have a univocal 

 series for y, belonging to the value of u chosen. Moreover, if u be real, so in this case are 

 u , u , ... , being obtained from equations of the first degree. 



Next, let there be 9 roots of equal value in \m = 0, one of which is taken. Striking out 

 the terms which certainly vanish, we have, to determine U, denoting 1 . 2 . 3 ... a by [a], 



(vu.a>"+ ...) + ( v 'u.w"+...)U+ ... +(p( e -^u.,v"+...)=j--y: + ^u.w m + ,.,),„ 



£7-6+1 



+ (^)u.ar + ...) j-^+ ... =0, 



where /u (8) « does not vanish. We proceed to examine the fractions 



(n + )-(« + ) (n +) - (n + ) (« + ) - m (n + ) - (m + ) 



1-0 ' 2-0 '" 6-0 (0+ l)-0 



in which the first (n + ) in the numerators means the same throughout. 



Here, since (n + ) - (m + ) is not greater than (n + ) - m, and since the denominators 

 increase, there must be a maximum at or before \(n+) - m\ : 9. Should it happen before, 

 a repetition of the process produces the same result, until a maximum at last has (n +) — m 

 in the numerator, though not the same (n +) with which we began. And by this time we 

 have obtained 9 solutions, all of positive degrees : for, in the general theorem, by the time 

 the maximum involves the use of k in the numerator, the number of solutions, after giving 

 each result due multiplicity, amounts to it - a. 



