CAUCHY'S PROOF OF TAYLOR'S THEOREM. 77 



and from this it follows of course that 



F(a+h)-F(a)_F'(a+0K) 



The reader who wishes to see the application of this 



result to the establishment of Taylor's Theorem, may pass 



on to Art. 106 at once, and then return to the consideration 



of the omitted Articles, in which we shall give another proof 



of the result, and also some geometrical illustrations. 



100. The enunciation of Art. 98 being supposed, we may 

 arrange the proof thus : 



Divide h into a number of equal parts, and let a denote 

 one of these parts. Consider the fractions 

 F(a+a)-F(a) F(a + 2oi}-F(a+a.} F(a+3a)-F(a + 2a) 

 /(a+a)-/(a)' /( a +2a)-/(a+a) ' /(a + 3a)-/(a+2a) ' 



F(a+h)-F(a+h-ai) 

 ~ f(a + h)-f(a + h-d)" 



Form a new fraction by adding together all the nume- 

 rators in (1) for a new numerator, and all the denominators 

 in (1) for a new denominator. We thus obtain 



-F(a) 



Since the denominators which occur in (1) have by hypo- 

 thesis all the same sign, we know from algebra that the 

 fraction (2) lies in value between the greatest and least of 

 those in (1). Now 



F(a -f a) - F(a) 



F(a + a) - F(a) = _ a _ 

 /(a + a) -/(a) == /(o+a)-/(a) ' 



a. 



W (fi\ 



if then we put this fraction equal to ., . + ft, we know 



/ ( a ) 

 that ft diminishes without limit when a does so. 



Similarly, 



F(a + 2a) - F(a + a) F' (a + a) 

 /(a+2a)-/(a+o) "/'( + ) ' 



