INCREMENTS. 





~6~ 



but a real quantity, pofit'iTC or negative, in other cafes, 

 which rauft be determined from the nature of the pro- 

 blem. This remark will be of confiderable importance in 

 what follows. 



lo. Let 



faciora 

 viz. 



us thet) confider thofe produfts, of which the 

 continually increafed by a conftant dilTerence ; 



:«) 



a) (.^ 



3^) 



-) ( 



A; 

 Ga. 



+ 2 a) 



tfi'Ax lla-.r" Ga^x I 



other fimilar quantities. 



And if it be required to find the integral of quantities of 

 the form 



x(x + A,) {x+2Ax) 

 x{x + ^x) (4;+2A.r) (jr + 3Ax) 

 &c. ike. &c. 

 Here it is obvious that the increment of 4? (.v + A*) ; or 

 I A(.v(.v^ A.r))=(.v + AA0(4r+2Ax)-..-(*4-Ax) i 

 whence, by multiplication and fubtra<^ion, we have 

 A (.V (.r + A a-)) - 2 A A- (* + A x). 



In the fame manner we iind, 

 A (x{x + A.t-) (.r + 2 A;r))=3A*'(.v + AiV>-+ 2A*); 



And thus again, 

 A ( T (.V + A x) (*■ 4- 2 A X) (.r + 3 A *)) = 4 A x 



(x + Ax) {x + 2 Ax) (xH-3 Ax) 

 and fo on of other fimilar produfts. Whence it appears, 

 that in order to find the increment of any produft of the 

 above form, we niuvt fupprefs the firft faftor x, and v/rite 

 in its place the increment A x, effected with a co-eificient 

 equal to the total number of faftors ; all the other part of 

 ; and fo 00 of the. expreflion remaining as before. 



Yv'hence again, converfely, the integral to any increment 

 of this form, wiU be found from the reverfe operation. 

 Thus for example, 



1. y 2 A X (.V + A .v) = .V (.f + A .r) 



2. /" 3 A .V (.r + A x) (.V + 2 A x) = X {x 4- A x) 

 (.v+ 2Ax) 



3. /'4 A .V (.V + A x) {X +ZA x) (t + 3 A^) = 

 .V (x'+ A .f) (x + 2 A.v) {x + S Ax) 

 and fo on of others. 



Whence, in order to find the correfponding integral to any 

 increment of the above form, which may be reprefented ge- 

 nerally by 



aAx{x-\-A x) (x + 2 A r) . . . (x -J- n A k), 

 wc mud change A .v in the firft faftor into x, and divide the 



whole by the number of faftors ; that is, 

 fa A X (x + A .v) (.V + 2 A .t) 

 a v ( V + A X) (;. -f- 2 A x) . 



Ax) 

 Ax). 



J 2A:(r 1 



the fum of which will be the increment required. 



And, in a fimilar manner, we may find the integral to any 

 other quantity of thefe forms. 



9. Rimark. — Before we proceed any further on this fiib- 

 jeft, it will be proper to attend to the correftion of irny in- 

 tegral, when from the nature of the problem under confjder- 

 ation fuch becomes neceffary. 



As the increments of any variable quantities x, and * + a> 

 are both exprcffed by Ax, the conftant part a having no ia- 

 crement, fo, reciprocally, the integral of the increment A x 

 may be x, or x + a ; therefore, wlien we have fo\ir.d the 

 integral of any increment, we muft add to it atoniiant quan- 

 tity, which will be zero, if the integral needi no correftion 



This part of the theory has been before confidered in the 

 preceding pages, in defcribing the method employed by M. 

 N'thaleof the Academy of Sciences, publiflied in 17 17. 



1 1. Let us now confider thofe fraftions, the denominators 

 of whicli are compofcd of faftors fimilar to thofe above de- 

 fcribed, T.'a. 



-i-Ax) 



:•) [X+ZAX) 



X {x + A 

 in which A x 



:) (x -I-.2A 

 s conftant. 



.) (.v. -I- 3 A;r); &c. 



