350 Mr. P. Nicholson on derivative Analysis. 



From the operation now executed we have the following 

 derivative equations, 



B,= a"B+A 



C,= 6"B,+B, 



D,= c"C,+C. 



&c. &c. 



Whence the law of derivation is obvious. 



Ex. 3. Divide the series C +- + ---+ -;;j- + 7^ + &c- 



(which is the quotient of the preceding example increased by 

 the quantity C) hy r- a" =s-b"'=t- d"=u-d:"=hc. 



Divisor. 



c+ 5- +^ + §+ ^>&c-fof ;=r ''■'='-'"="-'''^. 



c-'^ 



Quotient 



|£ . ^+^+-5l+&c. 



I r rs rst rstu 



rst 



&c. 

 From the operation now executed we have the following 

 derivative equations 



B,= a"C +B 

 C,= b'"B,-}-B, 

 D3=c"X3 + C: 

 &c. 

 Let Avwx &c. + Bwx &c. + Cx &c. + D &c. + &c. be any 

 function of u, and Av'w'x' &c. + B„T)'tD' &c. + C„v' &c. + D„ &c. 

 +&c. be another function of u equal to the former, then will 



C,=C+(w-t;')B, 

 C,= C,+(x-to')B, 

 &c. 



D,=D+(a— v')C, 

 &c. 



B, = B +{v-v')A 

 B,=B,+(w-to')A 

 B3=B,+(.r -x')A 



&c. 



And finally, let 71 be the greatest number of factors ; and if 

 c„ be the difference between the 7ith factors of the first term 

 of each series, b^ the difference between the w — 1th factors in 

 the second terms ; and so on, then will 



Bn = 



