536 Royal Irish Academy. 



linear functions of no more than (n — r) unknown quantities left to 

 be determined. 



. Upon this remarkable relation may be constructed a method well 

 adapted for the expeditious computation of numerical values of the 

 different derivees. 



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

 dary functions, 



a' . U — a V , 



b' .V -bV + n '.U 1 -aV„ 



c' . U - c . V + b' . U, - b V, + a' . U 2 - a V 2> &c. 



under the form of symmetric functions of the roots of the equations 

 U = 0, V = 0, by aid of the theorems developed in the London 

 and Edinburgh Philosophical Magazine, December 1839, and after- 

 wards proceeds to a more close examination of the final derivee re- 

 sulting 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) de- 

 pends. 



For instance, the value of the element, whose characteristics are r, 

 s, is the difference between two products : the one of the coefficient 

 rth in order occurring in the polynomial U, by that which comes sth 

 in order in V ; the other product is that of the coefficient sth. in 

 order of the polynomial V, by that rth in order of U ; so that if the 



degree of each equation be n, there will be altogether i — — — I such 



m 



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 

 (n — 4), till finally we reach a square of four blocks or of one, ac- 

 cording as n is even or odd. These flats are laid upon one another , 

 so as to form a regularly ascending pyramid, of which the two dia- 

 gonal planes are termed the planes of separation and symmetry re- 

 spectively. The former divides the pyramid into two halves, such 

 that no element on the one side of it is the same as that of any 

 block in the other. The plane of symmetry, as the name denotes, 

 divides the pyramid into two exactly similar parts ; it being a rule, 

 that all elements lying in any given line of a square {plafond) parallel 

 to the plane of separation are identical; moreover, the sum of the 

 characteristics is the same, for all elements lying anywhere in a plane 

 parallel to that of separation. 



All the terms in the final derivee are made up by multiplying 

 n elements of the pile together, under the sole restriction, that no 

 two or more terms of the said product shall lie in any one plane out 

 of the two sets of planes perpendicular to the sides of the squares. 

 The sign of any such product is determined by the places of either 

 set of planes parallel to a side of the squares and to one another, in 

 which the elements composing it may be conceived to lie. 



