g46 THIRD REPORT — 1833. 



^ ~ &c., which become equal to 1 when x = a or x =: b, &c., 

 X -^ o 



but which are ^ero for all other values of x, to show that the 



terms into which they are multiplied disappear from the deve^ 



lopement in all cases except for such specific values of x. 



The existence of the terms of the series for u is hypothetical 

 only, and the equation which must be satisfied, as the essential 

 condition of the existence of any assigned hypothetical term, at 

 once directs us to reject those terms which would lead to infi- 

 nite values of the differential coefficients, as well as those which 

 possess multiple values which are incompatible with those con- 

 tained in «/. It is quite obvious that upon no other principle 

 could we either reject such infinite values, or justify the con- 

 nexion of a series of terms with the general form of ?/, which 

 have no existence except for specific values of x. The con- 

 clusion obtained is of considerable importance, in as much as it 

 shows that the series of Taylor, if considered and investigated 

 as having a contingent, and not a necessary existence, may be 

 so exhibited as to comprehend all those cases in which it is 

 commonly said to fail : and it will thus enable us to bring under 

 the dominion of the differential calculus many peculiar cases in 

 its different applications which have hitherto required to be 

 treated by independent methods. 



Thus, if it was required to determine the value of the fraction 



(2 2\^ 



— " ^" , when x=sa,we should find it to be, 



x^ {x — ay 



fly 



. (x^ — a-f 



dx^ 



x^ {x — of 



or 



{x + af.-~-,'{x-nf 



d x^ 



a conclusion which would be justified by the developement of 

 the numerator and denominator of this fraction by the complete 

 form of Taylor's series, when x = a. 



Many delicate and rather obscure questions in the theory of 

 maxima and minima, particularly those which Euler has deno- 



