240 Prof. K. Pearson on certain Properties 



Hence, by the use of (7), (10), and (15), we deduce 



v^ = c 4 {n 4 + n 3 (15m 2 — 6) + rc 2 (25m 2 2 + 25m 1 m 2 — 65m 2 -4- 11) 



+ n(10w? 2 3 +30m 1 ?72 2 2 + 2()m 1 2 ?^ 2 — 75m 2 2 -65m 1 w 2 + 80m 2 --6) 

 4- m 2 + 6w 1 m 2 3 + llwi 1 2 ?w 2 2 + 6m l 3 m 2 — 16m 2 3 — 42wiW 2 2 

 — 24m 1 2 /?? 2 -f- S6m ] m 2 + 49w 2 2 — 24m 2 } 



-i-n(n-l)(n-2)(n-3). . . (18) 



But 2 



thus we find 



c 4 m 2 (n 2 + m 1 n-f w? 2 ) . , ,,., „ ., 



* = n'(»-l)(n-2)(n-V X {**+«>(*»» + *".*U 



+ rc 2 (3mim 2 + 6mi 2 + Qm 2 ) + n(3m 2 2 + lSm^m^ + 18m 2 2 }. (19) 

 Now a = — r, /3 =—qn ; 



m x r r\ Wo . 



n \ ' n> n 



Substitute these values in (19), and make n infinite. The 

 hypergeometrical series now becomes the binomial (p + q) r i 

 and we have 



/*t = c\l + 3(r-2)pq), 



a result already deduced (Phil. Trans, vol. clxxxvi. p. 317). 

 This serves to confirm (19). 



Dividing equation (19) by (1 — #), and putting = 1, we 

 find, by remembering that 



r r s)„=(^)„r{S}„"- te " , " : 



(" - 4) to) l => + 4m! + m 2 - 2) ( %4 ) x + (2mj + 4m 2 - 4) ( %3 ) x 

 + (6m 2 -2m!-l)( %2 ) 1+ (4m 2 -3m 1 +2)( %1 ) 1 



+ (/7? 2 -mi + l)(xo)i . . • 

 Whence 



c 



v 5 — —, tyt 577 577 rr { ft 5 + n 4 (3 lm 2 — 1 0) 



n(n— l)(w — 2)(n— 3)(n— 4) l v 



+ n 3 (90m 2 2 + 90m!m 2 — 220m 2 + 35) 



+ n 2 (65m 2 3 + 195m 1 m 2 2 + 130mi 2 m 2 - 485m 2 2 



-420m 1 m 2 + 535m 2 -50) 



*•» 



t 2 — uitJ//o.l7l 2 ""Oil 



+ 800m 2 2 -490m 2 + 24) + m 2 5 + 10miW 2 4 



-f 35m l 2 m 2 * + 50m l 3 m 2 2 + 24ra 1 4 m 2 — 160m 1 m 2 3 



- 256mi 2 m 2 2 -i20m 1 3 m 2 + 240™^ + 490/*!m 2 2 



-240m!m 2 - 30m 2 4 + 213tw 2 3 - 380m 3 2 + 120m 2 }, (20) 



+ n(15m 2 4 + 90??*im 2 3 +165m 1 2 m 2 2 + 90mi 3 m 2 



645m.m 2 2 — 370m 1 2 -m 2 -f 600m 1 m 2 



