CoTTEE — Method of Solving Legendreh and BesseVs Equations. 159 



and so on. The law of formation soon becomes apparent. Similarly, the 

 other particular solution is got by operating on Bof. It should be noted that, 

 in performing the integrations, we do not need to add any arbitrary constant, 

 as we are only looking for a particular solution ; and the final solution is 

 complete, since it contains two arbitrary constants. 



If in equation (2) we put k = ~ {n+1), the solution would appear in the 



form 



^ = ( 1 - ;/,)- 1 { Aor ("+!) + Bof } , (5) 



where 



The two operators are therefore equivalent. 



Equation (4) gives the solution in those cases in which 2?i is an odd 

 integer, as well as in other cases ; but the form of the series changes as soon 

 as an integration of or'^ has to be performed. As an example, let us take the 

 case in which 2n = -1. Here it appears as if the two particular integrals 

 become identical; but, on referring back to equation (3), we see that it 

 becomes, when multiplied by ^'"-("+^), 



.^^D-'x^D'y - ^By - ^x'hy = A'or\ 



so that the resulting integration is 



B'hr^D-hr'^BHj - vky = - AXogx - B, say ; 



or, y={l- x-'^Br^y-^B-HrWY^ (Ax-i logi'j + Bor^). 



Now the operator ^ = x'iB'^x'W-hrW- contains two integrations raising 

 the power of x by 2, two differentiations lowering the power by 2, and 

 multiplications which diminish the power by 2 ; therefore, the series is one 

 in descending powers of x differing by x"'. The expression x~i (A log x + B) 

 to be operated on will obviously give rise to a succession of terms of similar 



form ; for we have 



Bx' '" log x = X- ("' + 1) ( 1 - m log X), (6) 



B' ' X- "• log X = - r + log :/; • (7) 



^ m - 1 Vm - 1 ^ 



Let the result of the operation 



0'V'j"4 log X be (Ay + Br log x) x' 



4r + l 



where Ay and Br are the numerical coefficients whose law of formation is 

 to be determined. Operating once more by <^, and using the formulae (6) 

 and (7), we get finally 



4)M-5 



^'■ + 1 .f--i log X = (^,.+1 + i>V+i log x) x'~^ ' 



