133 



Upon this remarkable relation may be constructed a me- 

 thod well adapted for the expeditious computation of nume- 

 rical values of the diflerent derivees. 



He next, as a point of curiosity, exhibits the values of 

 the secondary functions 



a'. Uq—uFq, 



b'.Uo-bVo + a'.V\-aFu 



c'.Uo-c.Vo-\- b'. C/, - bV, -f a'. U., - aV^, 



&c. 

 under the form of symmetric functions of the roots of the 

 equations t/ = 0, V = 0, by aid of the theorems developed 

 in the " London and Edinburgh Philosophical Magazine," 

 December, 1839, and afterwards proceeds to a more close 

 examination of the final derivee resulting from two equations 

 each of the same (any given) degree. 



He conceives a number of cubic blocks each of which 

 has two numbers, termed its characteristics, inscribed upon 

 one of its faces, upon which the value of such a block (itself 

 called an element) depends. 



For instance, the value of the element, whose character- 

 istics are r, s, is the difference between two products : the 

 one of the coefficient r'* in order occurring in the polyno- 

 mial U, by that which comes «'* in order in V ; the other 

 product is that of the coefficient *'* in order of the polyno- 

 mial V, by that r'^ in order of U ; so that if the degree of 



each equation be n, there will be altogether ^^ — ——2. such 

 elements. 



The blocks are formed into squares or flats [plafonds) of 



which the number is - or — ^^ — according as n is even or 



odd. The first of these contains n blanks in a side, the 

 next [n — 2), the next (« — 4), till finally we reach a square 

 of four blocks or of one, according as n is even or odd. 

 These flats are laid upon one another so as to form a regu- 



