ƒ 



592 



Un U,n tUR = O 



for all values of m different from 71. 



Tlie development bv partial integration then becomes : 



ƒ 



where 



U2n R^"+^ da; = R^'" 7^1 — 2w i^^"-"- ff, + 2hi («— 1) i^2"-4 r; 

 (— l)''-i 2«-i n (n— 1) ... 2/2^ ( -1/' 2" . n.' y„+i 



^j =z j f72n-R dR, ff^ ■=. I yj22 dR etc. 



(11) 



and 



\U^2nR dR = {—ly> n\ icf„Rt 



If tiie limits are 1 and 0, then we have to put : 



ffn= CR^'^{R'—iy 

 so that 



U2n= (7A«i22"(i2' — 1)". 



Puttino; C=— we tind for the polvnomium : 



(2n)/ (2n— 1)/ (2n — 2)/ 



n! [n — 1)/ (n — 2).' 



where "6), denotes the /^'^' binomium-coefficient of the ?i^'^ power, 

 further : 

 1 11 



i U'2n R dR = 2" {2ny Cq,, R dR — (2/i) / p-^"+i {R"-—IY dR= — ^^^ . 







This new function maj be considered as a zonal harmonic general- 

 ized for the case of directed quantities and might be applied e.g. 

 to the distribution of hits on a target. 



The analogy of (12) with the zonal harmonic function becomes 



(2n— 1).' 



conspicuous it the latter (5), by multiplication by , be 



(n— ly 



given the form : 



Un = ^^ — ^ .t'" ' . ^^ ^"-2 + — . — — ^ ^n-A _ etc. 



{71 — 1)! 2 {n-2)! 2 {n—4.)! 



The expression (12) satisfies the differential equation: 



^(1-^^)^ + {l-'^R')'^-^ + hi{n^\)Rlhn=^. 

 an- dR 



