120 Mr MURPHY'S SECOND MEMOIR ON THE 



Hence, if we suppose the equation J,f{t) .^ = to hold true for an 

 indefinite number of entire values of x, the equation f{t) = will also 

 have an indefinitely great number of roots all lying between and 1, 

 and the curve, of which the ordinate is f{t), and the abscissa t, would 

 intersect that portion of the axis of x, of which the length is unity 

 measured from the origin in an indefinitely great number of points; 

 thus we have a property characteristic of this class of functions.* 



7. We have supposed J'{t) to consist of terms involving the 

 powers of t, but as we may proceed in like manner for any other 

 assumed form, we take the following as an example, because it leads 

 to some remarkable results. 



To find a rational function of h. 1. (f) as y(h. 1. t) of the lowest 

 possible dimensions, which may satisfy the equation ftf(h.\.t).t' = 0, 

 X being any integer from to n—1 inclusive. 



Put /(h. \.t) = \ + A, h.\.t+ A^ (h. 1. ff + + A„ (h. 1. t)', 



and observing that J,{h.\.{t)]"'.f = {-\f. •^^■^•:;\ , 



we get f,f{\,.l.t).t' = ^^-j^^^,+~^^^^- ± -(^^nyr.T-. 



and actually adding the fractions in the right-hand member of this 

 equation, the numerator which is a function of n dimensions, ought 

 to vanish when x is any number of the series 0, 1, 2...(w-l); that is, 



{x + 1)» -A,{x + !)"-■ + 1 . 2^2 (a; + 1)""' - 1 . 2 . 3 ^3 (a; + 1)""' 



= C.x.{x-\){x-^) {x-n->r\). 



Let Si represent the sum of the natural numbers 1, 2, 3.,..(«-l), n, 

 Si the sum of their products two by two, 

 ^^3 the sum of their products three by three, &c. 



* Vide Art. (4) in my first Memoir on the Inverse Method of Definite Integrals. 



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