Function of any Number of Variates. 

 Using this formula twice, we find, putting 



10 L = W 2 =....= W n = 0, 



449 



H (0,0,. ...0) 



*!*,.... *»+!.• ..&+!.. ..* 



= 7V,]J(0,0, ....0)-(-l) 



*l+*2 + *» r- 



^ 2 H 



KK 



h x k, 



or 



.l'n 



'dwsdwt 





& s +2....&» 



■*,+■ 



>»*, 



fa-(-l) 





I- (7) 



m *A 



*H-1... -.**+!.... *• 



rstmifa 



(-1) ^ [ ° ^A--*' 



1 



Hence we find, with the aid of formulae (5) and (6), 



• «Vi=0J 



»W60 



. O 



»»5io . . . 



. 



^420 . . 



. 



0*4110 .. 



. 



77*330 . . . 



. 



77*3210 .. 



.0 



W»31110 . . . 



. 



W2220 ... 



.0 



™22110 . . . 



. 



W1211U0.. 



. . (J 



™ 1111110 . 





- 15, 

 = 15?i2, 

 = 3-hl2r 12 2 , 

 = 3r23 + 12ri2ri3, 

 = 9ri2+6ri 2 3 , 

 = 3ri3 + 6ri2 7*23 + 6ri2 2 7*13, 



— 3ri3^24 + 3ri2r"34-|- 3r23ru+ 6ri2n4n3^ 

 = H-2ri2 2 + 2ri3 2 + 2r23 2 + 8ri2ri3r 2 3, 

 = ^34 + 2r 13 ri4 + 2r23 7*24 + 2ri2 2 r34 + 4ri 3 ri3 r24 



+ 4ri2ri4r23, 



I = ^23 7*45 + r 24 7*35 + 7'25 7*34 + 2ri2 713 7*45 + 2r 2 3 7*14 7*15 



+ 2ri 3 ri4r 2 5 + 2ri 3 ri57° 2 4 + 2ri2n4^35 + 2 712 7*15^4, 



= T23rursQ -f 7'13 7*247*66 + n2^34^56 + ^34^15^26 + 7*347*167*25 

 + ^24^15 7*36 + 7*24 7*16 7*35 + 7*147*25 7*36 + 7*147*26 7*35 

 + 7*23 7*15 7*46 + 7'23 7*16 7*45 + 7*137*257*46 + 7*13 7*26 7*45 

 + 7*12 7*35 7*46 + 7*12 7*36 7*45? 



and any other moment of the sixth order is obtained by 

 permutation of indices. 



Phil. Mac;. S. 6. Vol. 37 No. 220. April 1919. 2 I 



(8) 



