( 815 ) 



this might liave been expected as this vahie satisfies the condition (25), 

 which can be easily proved by successive partial integration. If we 

 put : /•„ =z 1, 



(fn — — = :— 2)'. Un «-•>■' 



and the expression for the .1-coefiicients becomes e(iual to tiiat given 



by Bki'ns. 



(/"^„ 

 Therefore — may be substituted for (fn for the same reasons as, 



instead of the ^^-functions, zonal iiarmonics might be employed; in 

 practice however no labour is saved by this substitution as then 

 the polynomia are charged with supertluons coefticients. After what 

 has been said in § 3 about a change of the scale value, it will l)e 

 sufficient to lemark that in this case also the great advantage which can 

 be derived from the inti'oduction of a scale factor is the adaptation 

 by means of the first term of the series to the shape of the curvp, 

 the surface remaining equal to unity. 



The equation of the curve then becomes : 



u = e-f^'-' \A^ l\ {h.v) + A, L\ {h.v) -f etc. . . (28) 



and : 



h etc.) 



n! V .\!{n~2)! ^ ^ 





(29) 



The choice of the scale factor is of course quite arbiti-ary, but, 

 in order to determine it in accordance with the nature of the curve, 

 it is desirable to put .l^ =: 0, then the average of the second order 

 can be used for the definition of h and it is easily seen that : 



_ 1 



The coeif\ of (29) in so far as they are independent of ;Mnay further 

 be omitted and written before (28), then the equation of the curve 

 becomes : 



« = ;'r e-''^'[A,U, + A,L\ + A,Cr^ + etc. 



If w^e take into consideration only the first term in the deve- 

 lopment, we find the exponential law in its simplest form as 



h 



u=r —r 



55- 



e' 



-A'x' 



