3H8 



I Un dx = (f\ 







X X 



1 U,i A-' clv = iv- ff,~2w y, -r- (fs ffs = r/a d.v, etc. 







By (2) it follows from these equations that the imposed conditions 

 are fultilled when, in the development 



X 



j Un .V" d.v — X" (p,—n A-"-l (p^... (— l)"-i n{n—\) . 27^,, ( — l)"n/ ^„+1 (3) 







(pn be given such a value that this function, as also its {n — 1) first 

 diiFerential-qnotients, become zero for .r = /i and ,i' = « and that then 



« a 



/7, = ^^ and ( fV c^.t- = (-1)" n/ ( ^„ ci.t- .... (4) 

 dxn J J 



This simple method of determining the terms of the required series 

 was indicated in 1833 bj Murphy as a new method of coming to 

 zonal harmonics ; in Thomson and Tait's "Natural Philosophy" it is 

 mentioned in article 782. 



The method, however, is by no means restricted to the calculation 

 of zonal harmonics but can easily be generalized and applied to other 

 circumstances than those mentioned above. 



Instead of a complete polynomium we can also consider separatelj^ 

 even and uneven polynomia ; polynomia multiplied by an exponential 

 factor as e~^'^ or e~^ may be used, and instead of dx we can take 

 xdx (plane) or x'^dx (space) as the element of integration, whereas 

 for X also quantities of another kind, e.g. sin a, may be substituted. 



3. If the limits are -j- 1 and — J , it is rational to put : 

 <fn = C{x' — iy C7,. =r=C-^(.t-- 1)« 



C being an arbitrary constant. 

 Putting 



