— . 



not convergent, a supplementary correction is considered to 

 be comprehended under the "&c." In tlie various transfor- 

 mations wiiicli this series is made to undergo, in order tliat it 

 may serve tiie purpose of the actual construction of logarithmic 

 tables, it will be found on examination that they are always 

 such as to preserve throughout the convergency of the series, 

 so that the correction adverted to disappears. If however we 

 rephice a by a — I, a being the base of the system, we then 

 render the series necessarily divergent; and it is common in 

 writings on this subject (see lor instance Miller's Diff. Calc, 

 p. 10) to apply to it in this state certain transformations, by 

 which it is said to be converted into a converging series. But 

 no diverging series can admit of such conversion; and whenever 

 this appears to be accomplished, it will always be found that 

 the original series is taken, not by itself, but in conjunction 

 with its correction ; and thus the change apparently brought 

 about independently of this correction, is, in reality, a new 

 development of the function generating the original series. 



There is a well known theorem of Lagrange, which, in the 

 case of Taylor's series, enables us to assign the limits within 

 which must lie the error we commit by taking any finite num- 

 ber of terms of the series as an equivalent for the undeveloped 

 function. In the form in which Lagrange delivered it, the 

 theorem is 



in which he says " u designe unequantite inconnue, mais ren- 

 fermee entre les limites et x " ( Theorie des Fonctions, p. 68) ; 

 and the same conditions are always said to be necessai-y when- 

 ever the theorem is announced. It is certainly of little or no 

 practical moment to correct this statement; yet in order that 

 extreme cases even may not be improperly excluded, it is 

 necessary to widen the limits so as actually to include and 

 X. For it is plain that if the series be finite, and we stop at 

 the last term, which it is conceivable we might sometimes do 

 without knowing that the final term v/as reached, ti would be 

 actually ; and if we stop at the first term, then in every case 

 u would be equal to x ; so that, leaving the term at which we 

 stop entirely unrestricted, the generality of the theorem re- 

 quires that the limits and x be included. From an exami- 

 nation of this theorem, I am inclined to think that it is capable 

 of greater definiteness and precision than is at present given 

 to it, the range between the limits depending in general upon 

 the place of the term at which we stop : but the discussion of 

 this point must be reserved for a future occasion. 



