434 PROFESSOR YOUNG, OX THE PRINCIPLE OF CONTINUITY, 



are all neutral. But, as already remarked, since the first of these is convergent, its sum does not 

 differ from that of the corresponding dependent series*. 



It is not in reference to series only that this distinction between neutrality and dependence has 

 been overlooked. It has been improperly neglected in the treatment of an extensive class of definite 

 integrals; all those, namely, that are analogous to periodic series, in respect to the indeterminate- 

 ness which they involve. It has already been shown what contrivance Poisson resorts to, in order 

 to get rid of this indeterminateness in the series : he destroys the indeterminateness of the integrals 

 by a similar artifice. The series were rendered determinate by multiplying their terms by the 

 ascending powers of a foreign factor ; thus bringing them under a law of continuity from which 

 originally they were wholly free. The integrals are rendered, in like manner, determinate by 

 introducing, under the sign of integration, a new variable: — an exponential multiplier, in virtue of 

 the variation of which, a bond of continuity is, as before, imposed upon the expression, and its 

 indeterminateness thus overruled. The following definite integrals quoted from Poisson, and those 

 who have espoused his principles, are all essentially indeterminate : — 



/» .00 ,00 /-ac 



rf.r sin rx, I dx cos rw, / rfa:,r""' sin rx, / drx" ' cos r.r, 



„ •'o •'" "^o 



/ dv .r" '■' sin ci-, / dx x" -cos rx, &c., &c., 



•^0 •'0 



the exponents of x in the last four being positive. A very little consideration will suffice to 

 convince us of this : we need only revert to the ordinary ideas involved in the method of quadratures: 

 for if in any of these forms the expression under the integral sign — omitting the dv — represent the 

 ordinate of a curve, we at once see that for x = eo — one of the proposed limits — that ordinate, and 

 therefore the area, or the entire integral, must be indeterminate. By introducing the factor e'"^', 

 for which there is of course not the slightest warranty, these forms become changed into the follow- 

 ing : — 



rf.r e""' sin r.r, / dxe~'"cosrx, / dx e''^'x'''' sin rx, 



•^it ^0 



/ dx e~"' d,"''^ cosrx, \ dx e'"' x'~'^ svnrx, / dxe''^' x"'- c(\s rx, 



•^a •'0 •'ci 



in reference to which the ordinates, at the limit x = co, all vanish, irrespective of the value of a. 

 If the integrations be now executed, each result will be a general expression involving o; and if we 

 seek what this expression becomes when a, by continuous variation, arrives at zero, we shall truly 

 obtain the limit of the integral ; that is to say, we shall obtain the last of the continuous series of 

 values which the integral passes through as a diminishes continuously, from some superior value, 

 down to zero. These results therefore are all valid, as limits of the changed integrals ; but have, 

 in strictness, nothing to do with the integrals originally proposed ; these latter being neutral, or 

 independent ; and therefore not included in the continuous series of values adverted to. 



The impossibility of reconciling some of the erroneous, but prevalent conclusions that have been 

 arrived at respecting the foregoing integrals, with certain known elementary truths, has led one or 

 two recent writers to pass too sweeping a condemnation on integrals of this kind ; and to reject, as 

 false, integrations that may easily be proved to be true. I shall advert to some of these presently. 

 But it may not be altogether out of place previously to remark, that much needless ingenuity seems 

 of late to have been expended in proving that sin od and cos co cannot be zero; although such 

 is unhesitatingly affirmed to be the case by the late Mr. Gregory f, and — with misgivings how- 



• See Note f B), al the eiul of this Paper. 



+ " Both the sine and the cosine of an infinite angle are equal to zero. '' Gregory's Ejamples, p. 477- 



