Calculus of Operations to Algebraical Expansions, 363 



which suppose equal to 



A„+B M V + C n V 2 + E„V 8 + ..., 

 the law of the coefficients of v heing the same as that before 

 given, using %'a, ^ ^'a, &c. for a, b, &c. With this variation, 

 we have, as before, 



'\jr^- 1 x=zyjro(. -{■ A^a, . X-\- ( A 2 ^"« + B 2 -v/r'«) . — 



This process has the further advantage of avoiding the diffi- 

 culties arising from ^having equal roots. If %x have m roots 

 equal to a, then 



( v V = / v V 



\2.3..m % V ^ / 



V \2.3..m % + 2.3..(m + l) % a -V-i-"7 



Forming the coefficients A w , B„, &c. with the new values of 

 0, b, c } &c, we have 



f % -^=>|r a + (A 1 + B 1 V + ..)V" (wl " 1) .D () .^'«.^ 



+ (A 2 + B 2 v + .0v- 2(Wi-1) -D • ^ -j +• • • 



J)2m-1 ^>2 



+ (A 2 + B 2 V+.0 2 .3 tt (2m~l) ^'2- f> -- 



\1.2..(m— 1) T 1.2..(m— 2) T / 



/A T* \ r 2 



V2.3..(2m-l) r + 2.3..(2m-2) r + 7 2^ 



It will be seen from the examples which have been given that 

 the object has been to exemplify the peculiar powers of the new 

 process rather than to exhibit any new results, though many of 

 those above given will be found to present themselves in a new 

 form. 



Dublin, August 22, 1853. 



2B2 



