the Differential and Integral Calculus. 33 



nitude in that firmament of science. Ever}' one must feel 

 that a notation employed by such authors must have some 

 peculiar advantages to recommend it ; still I believe neither 

 of these celebrated writers has given any reason for adopting 

 the suffix notation in preference to the common one. I wish 

 they had, but as the matter stands I can only adduce their 

 names in support of the f m mode. On the other hand I wish 

 to cite the reasons of two justly esteemed authors for employing 

 the old notation. 



Mr. Pratt, in the preface to his ' Principles of Mathematical 

 Philosophy,' ed. 1835, says, " The prevailing argument with 

 me for using the old differential and integral notation is the 

 excellence of Fourier's notation for definite integrals. I much 

 prefer that to any other that I have seen, and this naturally 

 led me back to the old form of differentials and integrals. 

 In case any of my readers are not acquainted with Fourier's 



notation, I now give it: / udx represents the integral of 

 »/ a 



the differential coefficient u, or of the differential udx, with 



respect to x taken between the limiting values a and b of x. 



In successive integrations the order of arrangement of the 



integrals is the same as that of the differentials : thus 



I I P dpdco represents the double integral of P with 



respect to /a. and w, the limits of [x being —1 and 1, and the 

 limits of :« being and 2?r." 



Mr. Gregory, in the preface to his ' Examples of the 

 Differential and Integral Calculus,' thus expresses himself: — 

 " I have adhered throughout to the notation of Leibnitz in 

 preference to that which has been of late revived and partially 

 adopted in this University. Of the differential notation I need 

 say nothing here, as it appears to be abandoned as an exclusive 

 system by those who introduced it ; but as the suffix notation 

 for integrals has been sanctioned by those whose names are 

 of high authority, I may state briefly some of my reasons for 

 differing from them. 



" In the first place, on considering the subject, I could find 

 no arguments against the use of the notation for differentials, 

 which did not apply with even greater force against that for 

 integrals; indeed, although there may be some cases in which 

 the use of the former is advantageous, I know of none in 

 which the latter does not appear to me to be inconvenient. 



" In the next place, I fully agree with Professor De Morgan 

 in an unwillingness to lose sight of the analogy to summation 

 which is implied in the old notation : and if it were at any 



Phil. Mae. S. 3. Vol. 24. No. 156. Jan. 1844. D 



