118 CELESTIAL MECHANICS. 1. ii. 8. 



ference or the fluxion of the variation, that is 



BAxzzA^x, and ^darzzd^o:. 



Supposing two successive values of the ordinate x, 

 corresponding' to the abscisses y and y + h, to be x and x 

 + Ax, and the curve or the equation to be so altered, that 

 the ordinates receive the addition characterized by 8 ; the 

 values corresponding to y and y-^h will then be x + dx, 

 and x-{-Ax + ^ (j7 + A^)or x-\-Ax-\-^x + dAx. If we now 

 suppose the latter variation to take place first, and then 

 the former, we shall have a; and x + ^x, both correspond- 

 ing to y ; and x + Ax and or + Sj; + A (^ + ^a;) -zzx-^-^x + Ax 

 -\-Adx, corresponding to y + h. Now the final effects 

 being the same, by the supposition, which ever change we 

 consider first, it follows that a' + Aa: + Sa-r§Aa7zzx + 3x + 

 Aa: + AS.r, and consequently SAr=A§j7, which is true 

 whatever be the magnitude of the changes, and conse- 

 quently when they become evanescent, so that we may 

 substitute d for A, and ddx will be still equal to 6^x, 



The truth of this propo- 

 sition may also be easily 

 understood from a geome- 

 trical illustration, the varia- 

 tion or difference, expressed 

 by A, arising from the com- 

 parison of the ordinates at 

 y and at y + h, and the 

 variation S relating to the 

 change produced in the 

 same ordinates by any arbitrary variation of the curve • 

 the same portion of the second ordinate representing ^Ax 

 and A^x. 



Scholium. The foundation of the method of indepen- 

 dent variations was laid by Leibnitz, under the name of 



.1/ + ^ 



