392 Mr. L. Isserlis : Application of Solid Hyper geometrical 

 variables where we know a priori that approximately 



Let us assume that cc'=l. Multiplying equations (21), 

 (22) we have 



(n-l)> 2 Qpo2 



1, 



or g(l— q) = — jy w ,, , . . . (69) 



■/(n — r)(n — r)rr 



(n-l)o-o-' 

 Now from (58) we have easily 



n—L 



Substituting fxT' "TXT ^ or r * r ' * n ^^ anc * ec l ua ^ n g 

 the two values of ^(1 — g) we find 



[4(n-l)+(9(n-2) 2 ]o-(7' = nV^/(?+l)^ + l). 



Or, since f rj — -^ , wis given by 



^.-D + ^V-^^^. . (71) 



NOW p = P\ijCF(J f . 



Hence if V* =/*(£+ l)(i| + 1), • • • • (72) 



n 2 (0-</>) + <4-4<9)+4<9-4 = O. . . (73) 



The rest of the solution follows as in §§7 and 8. It is to 

 be remarked that this solution applies to symmetrical dis- 

 tributions as well. For convenience in numerical applications 

 a table is appended, giving the various constants in the order 

 of calculation. 



§10. Numerical. — In numerical applications, we begin by 

 calculating the moments about some convenient origin and 

 transferring them to the mean by the formulae : — 



^02 = P'()2—P'01 2 , 



p03 =^'o3-3pVoi + Vol 3 . 



P0i = P'<)4— ±P'03P'01+$P'02P'01 2 — 3/oiS 



pot = v'm-Sp'oip'v + 1 Op Voi 2 — 10p' 02 pV + 4/ 01 4 , 

 Pii=p\i— p'oip'io, 



P21 = p'2i—p'2oP f oi—2p'iip'io + VioVoi, 

 and similar formulas obtained by interchanging the suffixes. 



