216 THIRD REPORT — 1833. 



Legendre, following closely in the footsteps of this illustrious 

 analyst, has succeeded in the investigation of methods by which 

 the values of this transcendent F (r) may be calculated to any 

 required degree of accuracy for all positive values of r, and has 



a"* x^ X" a-' 1 „ „ 



If we replace (« + 1)2 by a;2 + 2 a; + 1, we find 



d-^u X* '-^^ «:- r. . r. 



I^=r2 + T + T+C^+Ci. 



d-^ u 

 It is obvious that these two values of -, 5 cannot be made identical, 



without the aid of the proper arbitrary functions. 



dr u 

 (7.) Let u = f" where v =.f{x) : and let it be required to find TTr' 



d'' u 

 llie general expression for -j-^ , when r is a whole number, is generally 



extremely complicated, though the law of formation of its terms can always 

 be assigned. If the inexplicable expressions in the resulting series be re- 

 placed by their proper transcendents, the expression may be generalized for 

 any value of r. 



d V , .^d^ V , dr u 



li -J— =:p and if -j—^ = c, a constant quantity, then ~r^'=- n (n — 1) 



.... {n — r -\- I) v"-'' f 



c r (r — I) £_^ _i_ r (r— 1) (r — 2) (?• — 3) c^ v^ 7 



V + 1 (ra — r + 1) 'y^l.2.Qi — r+l) {n— r + 2)"^'^ ^H 



r (1 +w) „ r r (r — 1) cc , „ ) 



+ T(^ - --' + r(-;-i) --^ + ^- 



which is in a form adapted to any value of r. 



Tf _ ' 1 ■ 



dx^ 



+ 



X- x^ 



Rational functions of x may be resolved into a series of fractions, whose 

 denominators are of the form {x -f- a)", and whose numerators are constant 

 quantities, whose rth differential coefficients may be found by the methods 

 given above. Irrational functions must be treated by general methods similar 

 to that followed in the example just given, which will be more or less com- 

 plicated according to the greater or less number of successive simple differ- 

 entials of the function beneath the radical sign, which are not equal to zero. 



X 



