I\ REFERENCE TO CERTAIN RESULTS OF ANALYSIS. 437 



this omission, too, we further see how it happens that, in enquiries of this kind, we may be led from 

 premisses absolutely wrong, and that by a train of correct reasoning, to conclusions absolutely 

 right. We have only to take the results of differentiation here noticed, each with the indeterminate 

 constant suppressed, and which are thus erroneous, and to apply the reverse process of integration, in 

 onler to arrive at correct forms. Thus Poisson, starting from the false equation 



1 = cos - cos 20 + cos sy - cos 40 + he.*, 



in which he supposes Q< -n, multiplies by dO and integrates ; thus obtaining the true equation 



6 . ^ sin 2 sin .30 sin 40 



2 2 3 4 ^ ' 



and from a second integration, the other true equation 



TT &'■ ^ cos 2 cos 30 cos 40 



= cos + + &c. 



12 4 4 9 16 



Again : proceeding from the false equation, 



= sin - sin 30 + sin .50 - sin 70 + &c., 



he arrives, m a similar manner, at the true results 



TT cos 30 cos .5 cos 70 



- = COS0 - _- + -_ _ + &c. ... (5), 



7r0 . , sin 30 sin .50 



and — = sin + &c.: 



4 9 25 



in reference to which however, from neglecting the principle of continuity, he commits the error of 

 vU))posing {A) to fail when = tt, and (B) to fail when = — ; although, in virtue of that principle, 



both must necessarily holdj-. 



As supplementary to the foregoing observations on the principle of continuity, I would wish to 

 add a remark or two in reference to what has been called discontinuity : — a term which, I think, is 

 sometimes injudiciously employed in analysis. Many expressions called discontinuous, should rather 

 be considered as composed of different continuous groups united together under one general form. 

 Distinct continuities, .so to speak, may be comprehended in one and the same function ; and it is 

 obvious that these may bo separately discussed, and the aggregate of the entire group estimated, 

 without at all introducing the idea of discontinuity. For instance, certain functions, strbmitted to 

 integration, become infinite between assigned limits of x : — would it not be better, and indeed more- 

 accurate, to say, of such functions, that each consists of two continuous series of values, within the 

 proposed limits, both series terminating at the same absolute value of x, than to say that the 

 function becomes discontinuous for that value .'' To obtain the definite integral in such a case, we 

 should only have first to integrate over one of the continuous series of values, then to integrate over 

 the other continuous series, and to unite the results, taking special care that the terminal or initial 

 value of .X', which unites the two series, obeys the law of continuity impressed upon each. An<l in 



this way may the integration be correctly executed, however often infinity may occur betwein 



+ « 

 the proposed limits. The definite integrsil / .v'dx may serve for illustration. The function 



• That till* cquution is false, has already been shown by ihe 

 audior in the I'hil. Mafj. for December, 1114.1. Uut it is sufficient 

 to observe, both with respect to this ecjuation and that next 

 quoted, that it is inipoHsiblc, from the character of sin cc and 



I suppose Poisson considers the powers of his arbitrary multiplier. 

 *Mnfiniment peu diderenlc de runittS" to he virtually present in 

 these series, to destroy their |»criodic character. Mtil (his docs Udt 

 interfere with tlie principle in the text. 



eoi as. that the series. side of either can he a determinate (junntity. + .See Journal fl^ ."A'cn/c /*oh/ferfi. .Vnhicr win. pp. 'M'.i- 



