l)}j a Process of mere Inspection. 133 



1. (a). Attach (?« — l) zeros all to the right of the terms in 

 the {a) progi'ession : next attach (?« — 2) zeros to the right and 

 carry over to the left; next attach (m — 3) zeros to the right 

 and carry over 2 to the left. Proceed in like manner until 

 all the (m — 1) zeros are carried over to the left and none re- 

 main on the right. 



The {r)i) lines thus formed are to be written under one an- 

 other. 



1. (i) Proceed in like manner to form n lines out of the 

 (Z>) progression by scattei'ing (w— 1) zeros between the right 

 and left. 



2. If we write these («) lines under the iiii) lines last ob- 

 tained, we shall have a solid square (w + «) terms deep and 

 (?« + 7i) terms broad. 



3. Denote the lines of this square by arbitrary characters, 

 which write down in vertical order and permute in every 

 possible way, but separate the permutations that can be de- 

 rived from one another by an even number of interchanges 

 (effected between contiguous terms) from the rest ; there will 

 thus behalf of one kind and half of another. 



4. Now arrange the {m-\-7i) lines accordingly, so as to ob- 

 tain ^ {m + n . m + n — l 2.1) squares of one kind which 



shall be called positive squares, and an equal number of the 

 opposite kind which shall be called negative. 



Draw diagonals in the same direction in all the squares ; 

 multiply the coefficients that stand in any diagonal line together: 

 take the sum of the diagonal products of the positive squares, 

 and the sum of the diagonal products of the negative squares ; 

 the difference between these two sums is the prime deriva- 

 tive of the zero degree, i. e. is the result of elimination be- 

 tween the two given equations reduced to its ultimate state of 

 simplicity, there will be no irrelevant factors to reject, and no 

 terms which mutually destroy. 



Example. To eliminate between 



a x^ ■{■ li X -\- c = 

 I x^ + mx -\- n = 



I write down 



.(1) 

 .(2) 



■(3) 

 .(4) 



I permute the four characters (1) (2) (3) (4) distinguish- 

 mg them into positive and negative; thus I write together 



