70 Mb murphy, ON ELIMINATION BETWEEN AN INDEFINITE 



Another useful observation may be made in this place ; if the function 

 which represents the a;"' eqvxation were discontiiuious, i. e. if any of the 

 equations, for instance the second, were 



3 + 5 ' 7 ~ ' 



2 2 2 



and consequently an exception to the general law expressed by the x^^ 



equation, we should have then N—0 when x=\,S, 4 w, also when 



A' = ^, but not when a; = 2, hence in this case, 



iV=c. (ar-i)(x-l) {x-S) (x-4) {x-n); 



after this the remainder of the process would be the same as before. 



We have been thus particular about the preceding example, as being 

 well calculated to shew the spirit and advantages of the present method. 



The next class of equations, which may be solved by the first principle 

 alone, consists of those in which the terms composing the a;*'' equation 

 contain common factors ; for if we then assign to x such values as may 

 successively cause such factors to vanish, the unknown quantities will 

 be determined. 



Example : 



To find the values of asj, %2, %^ a, subject to the n equations 



following; viz. 



a:, + 1 ,2.S!2 + 1.2.3.S53+ + 1.2.3 «s!„= -1, 



2s!, + 2.3.a!2 + 2.3.4.X3+ + 2 . 3 . 4...(w + l)x„= -1, 



3a, + 3.4.&, + 3.4.5.«3 + + 3 . 4. 5...(w+2)a;„= -1, 



n8Sl + ?i(w + l)8:2+M(w + l)(w + 2)S83+ + W (w + 1) (« + 2)...2W2!„= -1. 



If we transpose the right-hand member of the above equations, the 

 .r"" or general equation becomes 



\ + x%,+ X {x + l)%.,^ X {x-\-\){x + ^) .%; + +a;(d;+l)(a:+2)...2x.!£„ = 0. 



