6 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
Thus 
.. (xviii.). 
Further, let us write {v n _^) x = k as v u _ and similarly (w„_ ] ) y = i as w„_ v 
= H. v,^ l5 W n = K. w n _ x .... 
We have then from (i.) 
I=££)>* 
1 r® co ( v n _ — 
= 7 ^ j j k e~' (x * +y2) dxdy + S (j- HK v n _ Y w n _ x 
(b + d)(c + d) ”(r» - — \ 
d - ^ HK V„_iW „_] j 
by (ii.) and (iii.). 
Or, remembering that N = a + b -f c + d, we can write this 
— 5c ~ (r n ~ - 
N 2 HK = ? (|F 
= r + £ hJc + f (P - 1) (P - 1) + ~A(A 9 - 3 )k{k* -3) 
+ l^ 4 - 6 ^ +3 )^ 4 “ 6p + 3 ) 
+ ~ h(W - 10P + 15) k (P - 10P + 15) 
+ r^(P - 15P + 45P - 15) (P - 15P + 45P- 15) 
+ — 21P + 105P - 105)&(P - 2lP + 105£ - 
Then 
and 
7 1 8 7 5 . 127 7 
« = XI + |y XI 3 + |-gXi 5 + -yy XI' + • ■ • 
1 ,7T-L 1 2 , 7 , , 127 . , 
"jj — d'- j7r ^1 + | 2 X 1 + p£ Xi + j g Xi + • 
7 1 3 7 5 127 , 
+ + l 5 X2 + ir & + - ■ - 
1 —-r 1 V2 7 ^ 127 
K " + ]l 1 + {4 X! + W X! 
These follow from the considerations that if 
Xi = X2 = 2 , 
dcf> 
~dh 
fIR 
d<f>. i 
- 1 = H, 
= - h. 
_ k 
dk ~ K ’ 
<ZK _ 
dfo ~ ’ 
105) + , &c. 
. . (xix.). 
whence it is easy to find the successive differentials of h with regard to 4>i and & with regard to and 
then obtain the above results by Maclaurin’s theorem. There is, of course, no difficulty in calculating 
H and Iv from (xvii.) directly. That method was adopted in the numerical illustrations. 
