133 



Upon this remarkable relation may be constructed a me- 

 thod well adapted for the expeditious computation of nume- 

 rical values of the different derivees. 



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

 the secondary functions 



a'.Uo— aFo, 



y. c/o - hv^ 4- «'■ ^'i - «^i> 



c'. CTo - c. Fo + h'. U, - hV, + a'. U., - aF,, 

 &c. 

 under the form of symmetric functions of the roots of the 

 equations C/ = 0, F= 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, insci'ibed 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 i-"' in order occurring in the polyno- 

 mial U, by that which comes s"' in order in F,- the other 

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

 mial V, by that /•"* in order of U ; so that if the degree of 



each equation be n, there will be altogether ^^ 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 (ra — 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- 



