NOTICED BY SIR WILLIAM HAMILTON. - 321 



Dem. As lim ,=o {y(A")| is an infinitesimal of the finite and positive 



fix) 

 order =^, it follows that the limit of the ratio ■^-^— ' for the values of .v 



x" 



which converge indefinitely towards zero, is equal to a finite quantity. 

 This finite quantity is the coeflScient A. From this and from the con- 

 tinuity of the functions (the difference of two continuous functions 

 being itself a continuous function) it follows that if we make x increase 

 insensibly from zero to some finite quantity within the limit .r, the 

 values which the functions 



successively assume, may be respectively represented by Bxi^, Cxy, Dx' 

 where B, C, D are finite, and a, /3, 7, ^ finite and positive with the 

 law j3>a, 7>/3, ^>7. Therefore, reducing and transposing, we .see that 

 for any finite value of x within the limit x^, /{x) may be analytically 

 represented by the series Ax" + Bxl^+ Cxy + Ux^ + lac. q.e.d. 



Cor. We may from the preceding proposition deduce a mode of 

 finding successively the terms Ax", Bx^, Cx^, and thus of actually 

 effecting the developement oi f{x). This is best explained by an example. 

 I^et y(.j-) be sin a; and assume smx = Ax + B xl^ + Cx^ + 6ie. Dividing 



Sin X 

 by x" we get — ~ = A + Bx'^-" + Cxy-" + ScC. Now making x con- 

 verge indefinitely towards zero, as /3>a and y>a, it is manifest that 



(Sin ^\ 

 — ^ j where a is that finite and positive number 



/Sin ^\ 

 which can render lini^^o f— ^^j a finite quantity. But by the ordinary 



rules of the Differential Calculus for finding the values of fractions which 



for certain values of the variable become -, we find that 



,. /sm x\ ,. / coso; 



hm.,=„ — —] = hm„o , 



\ X" I \aX 



a-l 



which for a = 1 becomes finite and equal to unity. Therefore 



sin .r = X + BxP + Cxy + &c. 



