Eleanor Pairman and Karl Pearson 
235 
Now all higher differentials of Z than the fifth vanish, and therefore we may 
(PZ' d'' Z' 
cancel the first two terms of and the first four of -r-- 
The value of 
,d?Z 
starts with the term 462s (s — l){s — 2) (s — 3) (s — 4) (s — 5) af^ and accordingly 
this and all terras beyond vanish for s = 0, 1, ... 5, or this 11th differential of Z' is 
zero for our purposes. 
We have now to give s in succession the values 0 to 5, and subtract the result- 
for the first from those for the second terminal : 
0 
s = 2 
s = 3: 
dZ^ 
da; 
d'Z' 
dx^ 
dZ' ' 
dx _ 
d'Z' 
dx^ 
'dJ>Z'' 
daf 
dZ^' 
dx 
d'Z' 
dx^ 
d^Z'' 
dx^ 
d'Z' 
' dZ' ~ 
dx 
'd'Z' 
_ dx^ 
d'Z' 
dx^ 
d'Z' 
dx' 
hp 
hp' hp 
h X 
ho" h„ 
I Xp ./.fl 
Y h~^ 
lip lip 
"d'Z'l 
dj? _ 
:< 
'd'Z' " 
dx' 
+ 5 f-^ - - 
2,x„ 
_ A 
h 
Hp 
'd}'Z'~ 
h 
■^ = 0; 
N, 
N, 
+ 6 
h p hp 
hn Jin 
\m+'Z'' 
da?'i+^ 
:0 if q>2; 
+ 
lip'' lip- hu- /(„-; 
&4 Xp 
= 42 
a-. 
N, 
'd^^Z^ 
^da;"i+'^ 
= 0, 
- 20 
q > 3 
hoJ 
b. 
+ 
hn' 
\ hp //o 
Xp" Xif \ 
hp ho'/ 
hp^ lln' 
+ 60 
h _d 
hp- ho". 
- 126 
hp' hp 
^1 + 210 
ho' ho 
(h 
«4 
V/'/ h,\ 
d-'i+'Z' 
\lx"i+' 
= 0, (/ > 3 ; 
