256 



DEVELOPMENT of a certain 



+ AB") 



— 2C 



-f- in A J 



— nC J 



+ 4 AE ] 

 -3D 



— cF >fin4<p&c. = 0; 



+ 2 ACt +.3AD1 



— B J — 2C 

 }>fin<p— 3D j>fin2<p — 4 E lfm^(p— 5F 



+ »B + aC + «D 



— ;/D J — «E J — »F J 

 hence we readily derive the following feries of equations 



c = 



D = 

 E = 



F- 



« + 2 



2A 



3^ 



7; + 4 



4 A 



B + 

 C + 

 D-f- 



n + 2 

 » — I 



» + 3 

 « — 2 

 « + 4 



A 

 B 

 G 



E-f-^D 



n+ $ ' n + $ 

 &C. &C 



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

 cients of the development of the formula (a 2 -{- b 2 — lab cof <p)„, 

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

 can alfo determine all the coefficients of the development of 

 (a 2 -j-b 2 — iab cof^p)' 2-1 , where the exponent is n — 1, by means 

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

 termine the coefficients when the exponent is n+q where q de- 

 notes any whole number whatfoever *. 

 For let us affume 

 (a 2 + b 2 — 2ab cof (p)"- 1 zA'-fB' cof <p -f- C' cof 2^ -f- D' cof 39 + &c. 

 Then, 



C=(a 2 +b*~- 2^cofp)(A'-fB'cof<p-f-C'cof2<p+&c.) 



W-**bcoBpy^ A + B co{(p + c cof 2 ^ D cof ^ + &c: 



From thefe two values of (tf + b 1 — 2^ cof <p)% by due redudion, 



and 



* Traite du Calcul Differentiel et du Calcul Integral, par Lacroix ; vol. ii» 

 page X20t 



