272 THIRD REPORT — 1833. 



rating functions. For the same principle would justify us in 

 rejecting I'emainders after an infinite number of tei'ms, whatever 

 their arithmetical values may be ; for such remainders can in- 

 fluence no terms at a finite distance from the origin, and there- 

 fore can in no respect afl^ect any reverse operation, by which 

 it may be required to pass from the series to any expression 

 dependent upon the generating function. Thus, if 



a o o 



=.a + ax + ax^-\- &c = *, 



1 — .r 



we shall get 



a = {\ — x) s = a, 



if we reject remainders after an infinite number of terms ; and 

 similarly in other cases. It would thus appear that algebraical 

 equivalence is not necessarily dependent upon the arithmetical 

 equality of the series and its generating function. 



It is, however, an inquiry of the utmost importance to be able 

 to ascertain when this arithmetical equality exists; or, in other 

 words, to ascertain under what circumstances we can determine 

 the sum of the series, either from our knowledge of the law of 

 formation of its successive terms, or approximate, to any required 



a, z'' + 02 ^''■"' ->r . . a^z 



z"- 1 



which becomes by the application of the ordinary rule of the differential cal- 

 culus, when » = 1 or .r = 1, 



p Oi + (p — 1) Qo + . . a^ 

 p ' 



which is the average or mean value determined by Bernoulli's rule. 



The discussion of the values of these periodic series has-been resumed by 

 Poisson in the twelfth volume of the Journal de I'Ecole Polytecknique. He 

 considers them as the limits of these series when considered as converging 

 series, a view^ of their origin and meaning which is almost entirely coincident 

 with that of Lagrange. Thus, the sum of the series 



sin 5 -|- p sin (.t + q) + p^ sin (2 x + q) + &c. 



is equal to 



sin q + p sin (x — 5) 



1 — 2 p cos X + 2->" 



•whenp is less than 1, an expression which degenerates, when p = 1, into 



11 X 



— sin 5- -I- — cos q cot — . 



which may be considered, therefore, as the limit of the sum of the series 

 sin q + sin (.r + q) + sin (2 x + q) + &c. in infin. 



