68 Mr murphy, ON ELIMINATION BETWEEN AN INDEFINITE 



Hence we have 7^ =0, and therefore iV=0, provided a; is any num- 

 ber of the series 1, 2, 3,....ti and consequently iV (which is of ?i dimensions) 



has a factor (^ — 1) U — 2) (x — ti); and can therefore admit of no 



other factor, but a constant c. 



Hence we have in general, 



^^ X x + 1 x + 2 x+n x(x+l){x + 2) {x + 3)...{x+n)' 



Multiply this equation by x, and then put x = 0, hence c = ( — 1)\ 



Multiply the same by x + 1, and then put x = —I; 



, n n + 1 



hence ssi = - - . — - — . 



Similarly, multiply by x + 2, and put x= —2, 



_ n.{tt—l) {n + !)(« + 2) 

 • • '^ ~ 1 . 2 • 1 . 2 ' 



and generally, if we multiply equation (a) by x + m, and then put 

 x= -m, we get 



'"'^ ' 'I. 2.3 m ' 1.2 m ' 



It is clear from this example, that if the general or x^^ equation were 



a+bx^ a' + h'x ^ d'\¥x ^ «<"' + i<"'x ~ "' 



we should find the sum of the fractions composing the left-hand member 

 to be 



c .{x—\){x — 2) {x — n) 



(a + hx) [a + h'x) («" + h"x) (a" + i^a;) ' 



then multiplying by n + bx and putting x= — j, we should find e, 



-I 

 multiplying by a'+b'x and putting .r=-,,, we should find s,, 



&c &c &c 



