438 Mr. P. Nicholson on derivative Analysis. 



10 -3 



4326 

 And this is the same example as was published in the post- 

 script to my Tract on Involution and Evolution, and it is the 

 same as the operation now detailed, except that the first, 

 third, and fifth columns are omitted. 



rr- A A B, C, D, , o 

 It — = — + — -^ + — - H + &c. 



k r rs TSt rslu 



* 1 -f. A , B B B. Ca , Da , Q 



And if ^ + ■!-= — + — + -7 + -- +&C. 



Ml r rs rst rstii 



Then will B,= B{r-l)i-A,C. = B.,{s-l) + T>„T>,= C,{t-l) 

 + C, &c. 

 That is, the coefficient of any term n is equal to the j^roduct 

 of the coefficient of the next preceding term ?i— 1, and a cer- 

 tain factor plus the coefficient of the corresponding term ?i — l 

 of the preceding series. 



This other factor is the remainder which arises by sub- 

 tracting the last factor of any term of the first side of the se- 

 cond series from the last factor of the (u — l)th term of the 

 second side. 



For let )■ — k = a' r — I =a" r—m=-a"' 

 s-k = b' s-l =h" s —m = h"' 

 t — k = c' t — I =c" t —m = c" 



Then will k =zr — a' = s — b' z=. t — c' &c. 

 I =r — a" =s — b" =:t — c" &c. 

 m=r — a'" = 5 — b'" = t — d" &c. 

 &c. 



Divide the first side of the equation A = A by k, and the 

 second side by its equals r—a\ s — b', t—c', u—d' &c. by ex- 

 ample 1 .division, page 1 0, and we have 



4 =£+A+£^+^+&c. 



* r rs rst rslu 



To 



