hy a Process of mere Inspection. 135 



give as the sum of the diagonal products 



lbmc-\-alnc-\-a7nbn-\-lacn 

 be i.elbmc-\-ainhn-{-2acln. 



Thus the result of eliminating between ax^^+bx+c^O 



I x^ +mx -\- a = 

 ought to, and is 

 a- n^ + P c^ — 2 a c In + Ib^ n + atn^c — lbmc--a m bn ■=■ 0. 



BmU for finding the 'prime derivative of tlie \st degree^ lahicJi is 

 of the form Aa; — B. 



1. Begin as before, only attach one zero less to each pro- 

 gression ; we shall thus obtain not a square, but an oblong- 

 broader than it is deep, containing (???+w — 2) rows, and 

 (m + ?i — 1) terms in each row: in a word, (m + w — 2) rows, 

 and {in-\-n — \) columns. 



To find (A) reject the column at the extreme right, we 

 thus recover a square arrangement (m + w — 2) terms, broad 

 and deep. 



Proceed with this new square as with the former one ; the 

 difference between the sums of the positive and negative diago- 

 nal products will give A. 



To find B, do just the same thing, with the exception of 

 striking off not the last column, but the last but one. 



"Rule for finding the prime derivative of any degree^ say the 

 rth, viz. A^. x"" — A,— i x'-^ + + Aq . 



Begin with adding zeros as before, but the number to be 

 added to the (a) progression is (m — r) and to the {b) pro- 

 gression (« — r). 



There will thus be formed an oblong containing {jn + ti — 2r) 

 rows, and (m + n — r) terms in each row, and therefore the 

 same number of columns. 



To find any coefficient as Ag , strike off all the last (/'+ 1) 

 columns except that which is (s) places distant from the ex- 

 treme right, and proceed with the resulting squares as before. 



Through the well-known ingenuity and kindly proferred 

 help of a distinguished friend, 1 trust to be able to get a ma- 

 chine made for working Sturm's theorem, and indeed all pro- 

 blems of derivation, after the method here expounded ; on 

 which subject I have a great deal more yet to say, than can 

 be inferred from this or my preceding papers. 



University College, London, Jan. 16, 1840. 



[To be continued.] 



