278 
PEOFESSOK K. PEAESON ON THE MATHEMATICAL THEOEY 
Hence, since and /3o can always be found, we have general expressions for d 
and Sk. in the case of the first four moments being arbitrary. 
Further, if we increase our series and write :— 
/^i = /^3V/^2^ A = 
^4 = P' 6 ^ 2 ^ A = y®6 = .... (xxxh.), 
we have the following perfectly general results, the law of frequency being any 
whatever:— 
( 4 / 34 , — 24^63 + 36 + 9 ySi ^3 ~ 12/83 + 35 ^ 1 ).(xxxiii.). 
ntp; = ^6 — + 4 / 33 ^ — + 16/30^1 — 8 ^ 3 + 16/81 .(xxxiv.). 
= 2^5 — 3 ^ 4 ^! — 4 ^ 3^2 -b 6^.2^/3^ + 3 ^ 1/33 — 6/33 + 1 2/3^2 + 24^i . (xxxv.). 
Lastly if K = 6 + — 2 ^..(xxxvi.). 
= il3, - 16/3 ,+ 7213,13^ - 24ft - 72/3,/3/ + 48ftft + 81^,=ft 
- lOSftft-4ft® - 188ftft +72ft+ 17]ft‘>+100ft . . . . .(xxxvii.). 
Thus the probable errors ‘67449 ‘67449 ‘67449 Sk of the quantities /3i, 
and what I have termed the criterion K, can be found whatever he the laiv of 
frequency. 
Let = H-zliHy, fhen 
= 2 V\/A.(xxxviii.), 
and its value can be found from (xxxiii.). Knowing S^„, and the correlation of 
errors in and ^ 83 , i.e., we can find at once the probable errors in d and Sk. 
from (xxx.) and (xxxi.). Thus with very great generality as to the law of frequency, 
we can test how far the distribution is a random sample from a population following 
any law whatever. 
Applying the above general results to the special case of the normal curve, we find 
siriC 0 
ft = 0 . ft = 3 , ft = 0 . ft = 15, ft = 0, ft = 105, 
Probable error of /3, = 0 
/3o = ‘67449 X a/^ . . . 
(xxxix.). 
v/ft" = -67449 X f ^ . 
.(xl.), 
= 0 
K = ‘67449 a/~ 
V n 
.(xli-f, 
d = ‘67449 a/ - o- 
V 2?t 
. 
Sk. = ‘67449 a/— 
V 2n 
. 
