double, treble, and other Systems of Algebraic Equations. 427 



torial irrelevancy, but also of linear redundancy, which latter 

 terra I use to signify the reappearance of the same combination 

 of prefixes, sometimes with positive and sometimes with nega- 

 tive signs : furthermore, it follows obviously from the nature 

 of the pi'ocess that no numerical quantity in the final derivee 

 will be greater than the higher of the indices of the two given 

 polynomials. 



Part II. — Ternary Systems. 



Case A. — Indices all equal. 



Method I. 



Let there be now three proposees, U, V, W, integer com- 

 plete homogeneous functions of x, y, 2, each of the degree n : 



let r + r'^ r"= n—\, s + s' + s" = ?^— I, t + t' -\- t"= n — 1, 



aT ./ . z"-" . u, ^ .y' . /' . V, .r' .y' . /' . W 



will, as above, be called the augmentees of U, V, W, and 

 every other part of the notation previously described is to be 

 preserved. 



Suppose now U = 0, V = 0, W = 0, 



we shall have as many augmentative equations formed from 

 each proposee as there are ways of stowing away n things be- 

 tween three lockers (vacancies admissible)*, i. e. n . — - — of 



each kind; in all, therefore, 3. —^—- , and every one of 



these will be of the degree In — \, so that the number of 



arguments to be eliminated is equal to the number of ways of 



stowing away 2 ?i — 1 things between three lockers (empty 



. 2 « . (2 n + 1) 

 ones counting), 1. e. ^—- -, 



As yet, then, we have not enough equations for eliminating 

 these linearly. 



Make, however, ci + ^ + y = n+ 1, 



and write U = x" .F + / .F + z'>' . F" = 0, 



V = x" . G + / . G' + a''. G" = 0, 



W = ^^ . H + / . H' + ^''. H" = 0, 



it will always be possible to make the multipliers of x', y, z'*' 

 integer functions: for if we look to any argument in U, V, or 

 W, it is of the form x" .3/* . s*^, and one of the letters a, b, c 

 must be not less than its correspondent «, /3, y, for otherwise 

 a -\- b + c would be not greater than « + /3 + y — 3, i. e. 

 n would be not greater than {n -\- 1) — 3, or » — 2, which is 



* See for Latin translation the preceding note. 



