Demonstration 0/ Newton's Rule. 871 



argument I have adopted to be very similar to that which New- 

 ton himself must have employed : I trust that there is no unpar- 

 donable presumption in my thus thinking. Newton and his 

 contemporaries— more especially Maclaurin — paid much atten- 

 tion to, and made much use, in their algebraical investigations, 

 of, limiting equations, of which they gave many varieties of form. 

 It is not likely that Newton had anticipated any of the peculiar 

 views of Fourier ; it is more probable that he arrived at his Rule, 

 as I have arrived at it, solely from the consideration of limiting 

 equations, and that his investigation of it was suppressed on 

 account of its length. In reference to his researches on the 

 Doctrine of Equations, in his ( Universal Arithmetic/ he says 

 (Raphson's Translation, 1720, p. 227), " Hitherto I have shown 

 the properties, transmutations, and reductions of all sorts of equa- 

 tions. I have not always joined the demonstrations, because 

 they seemed too easy to need it, and sometimes cannot be laid 

 down without too much tediousness." 



In reference to the series of limiting equations marked (I) 

 below, I may here mention a property which I believe has not 

 previously been observed : — 



If the middle one of any three of the consecutive functions be 

 squared, the first two terms of the result multiplied by the pro- 

 per Newtonian factor (as suggested by the degree of the polyno- 

 mial), will always be the same as the first two terms in the pro- 

 duct of the extremes multiplied by the corresponding Newtonian 

 factor. 



This general property may be demonstrated as follows : — 



Let any consecutive three of the functions 



/w, m> \h*)>. 2V3W' &c (t) 



be represented by 

 AjsP + A!xP- l + 



-pAxP- 1 -^ -(p — l)A'xP- 2 + 



m m ' 



m(m+l) m(m + l) v ^ M>r ' 



The first two terms of the product of the first and third of these 

 expressions, retaining coefficients only, are 



or 



, * 1 J ?(ff-l)A 2 + - , 2 , 1N (j>-l) 8 AA'. . (1) 

 »i(wi+l) ^ m(m-\-l) Kr ' v ' 



