336 A Theory of Fluxions. 



11 S_ 



The fluent NWr=?^l -JLl -, &c.,and the fluent iYm 

 3 i 



5a 2 



1 5. 



O-J-l 2 ft 2 t\-\ 2 <y* 2 



= — — , &c. If the fluents taken are proportion- 



3a% 5a% 



11 -L 



al to their corresponding fluxions, we have, a 3 L — - : 



3 bai 



•L L 1 * £ 



< 2,n 2 x 2 n 2 x 2 . . i \ x 2 x* n z x^x' n 2 x 2 x' 



3a§~ ~5af ' ' <2a\ ' a% 2af 



Now by multiplying we have the product of the extremes 

 equal to the product of the means, that is, 



2n 2 x 2 x* 8n 2 x 3 x", n 2 x i x' _J2n 2 x 2 x' 8n 2 x 3 x* n 2 x 4 x* 



3a 15a 2 10a 3 3a 15ft 2 10ft 3 ' 



Whence it is evident, that proportional fluents have their 

 fluxions in the same ratio with themselves. 



4. In the variable quantity ax — x 2 , let the value of x pass 

 through every assignable magnitude from to ft, given one ; 

 and let the fluents taken be ax — a: 2 , A ax — Ax 2 , B ax — Bx 2 , 

 Cax—Cx 2 ; these will constitute a series of fluents, which 

 are proportional to their fluxions. For, selecting any two 

 fluents as A ax — Ax 2 , and Cftx — Cx 2 , their corresponding 

 fluxions are Aax — 2Axx% and Cax' — 2Cxx\ Suppose that 

 Aax -Ax 2 : Cax — Cx 2 ::Aax*-2Axx- : Cax- — 2Cxx\ By 

 multiplying, we obtain the identical equation, ACa 2 xx* — 

 3ACax 2 x-+2ACx 3 x-=ACa 2 xx--3ACax 2 x-+2ACx 3 x-,that 

 is, the product of the extremes is equal to the product of the 

 means. 



By a series or set of fluents or fluxions is to be understood 

 those that are in geometrical proportion, for it may be re- 

 marked " that two fluents cannot be of the same set, unless 

 they are of a nature to be compared with each other, that is, 

 the one must be a multiple of the other by a whole number, 

 or a fraction. And this multiple of the fluent, being con- 

 stant, will always be a multiple of the corresponding fluxion : 

 therefore the fluxion will vary as the fluent. Generally, x n : 

 nx n ~ 1 x".'.2y n : 2ny n ~ 1 y. If y=ax, thenx" : wx n_1 x*: :2(a"x n ) 

 : 2w(a n x"~ 1 )x- Multiplying the extremes and means, In 

 (a"x 2 ' l-1 )x•=2?^(a n x 2 ' l_1 )x\ ,1 Hence we derive the two fol- 



