Mr. P. Nicholson on derivative Analj/sis. 439 



To each side of this equation add B, and 

 ^ + B =B+A + ^ + £:+^ + &c. 



I; r rs rst rstu 



Divide the fii'st side of this equation by I, and each term of 

 the second side respectively by each of the equals }-—a", 

 s — h"', t—c", u—d'" &c. by example 2 division, page 11, and 



i:+l = ^+-%+-^+^+&c. 



kl I r rs rst rstu 



To each side of this equation add C, and 

 _-V+±+C ... =C+ -?-+^+-^+-^+&c. 



kl I r rs ' rst rstu 



Divide the first side of this equation by m, and each step of 

 the second side respectively by each of its equals r—a"\ 

 s — h'", t—c", u — d'" &c., and 



-A + ^ + _c =^+^^ + £i+^+&c. 



klm Im TO r rs rst rstu 



"Without proceeding further from the first division of the 

 equation by k and its equals, we have 



See the operation ex. 1 division, 

 page 10. 

 &c. 

 From die second division of the equation by I and its 

 equals, we have 



B, = «"B +A ) 



C, = 6"B , + B , > See the operation ex. 2 division, 

 D'=c"C:+C,J page 11. 



&c. 

 From the third division of the equation by m and its equals, 

 we have B,=a" C +B "i 



C3=6"'Bj+B, V See the operation ex. division. 

 D,=c"C,+ Cj 

 &c. 

 Aiul so on. Now by arranging these in columns, according 

 to the orders of the derived equals, we have 



B,=«' A I C. = // B, 



B-«''B + A C.,=^'"B, + B, 



B;=«'"C + B C=b'''B,+B, 



&c. &c. 



D,=c' C, 



D -c" C,+ C, 



, &c. 



rhe application of this theorem may be seen in my 

 binatorial Analysis, under the article Binomial Factors. It 

 a])plies to fractional infuiite series in die same manner as the 

 })receding theorem, which has been sufficiently illustrated by 

 e\:uii))U's, does to whole numbers. 



' XCI. On 



&c. 

 Com- 



