256 DEFELOP MENT of a certain 



+ 4AEI 

 -3D 



fin 31P — 5F 



+ nT> 



— «F J 

 hence we readily derive the following feries of equations : 



+ 3AD-, 

 j — B j — 2C 



«D J Tit. 



>fin4<p&c. = 0; 



« + 2 ?; + 2 



n+3 "+3 



■*" 7; +4 '^« + 4 



F = 



4A 



D 



&C- &C. 



5. But befides being able to determine the remaining coeffi- 

 cients of the development of the formula {a^ + ^' — 2^3 cof ^),„ 

 •where the exponent is n, by means of the firft two A and B, we 

 can alfo determine all the coefficients of the development of 

 («* + ^' — 2(ib co{'^)"~', where the exponent is « — i, by means 

 of the fame two coefficients A and B ; and by them we can de- 

 termine the coefficients when the exponent is wij', where q de- 

 notes any whole number whatfoever *. 

 For let us aflume 

 (a' + b''— 2ab cof (p)'-' = A' + B' cof (p-f C'cof 2(p + D' cof 3(p + &c. 

 Then, 



rn(^ ' +3 »— 2^3cof^)( A'+B'cof<p+C'cof2^&c.) 



('''+^'-""^^°^^^|= A + B cof ^ -f C cof 2? -f^ D cof 3? + &c: 



From thefe two values of (<?-+^' 



-2aZ'cof^)", by due reducflion, 



arid 



* Traite du Calcul Differentiel et du Calcul Integral, par Lacroix ; vol. ji- 

 page 120. 



