166 On Expansions in Tetrachoric Functions 
and 
r^-'e-^ ^ 1 V4! 1 V5! 
, = i (1 + Of,) T.. T4 
_ ^_ _ \/7 ! /j j + :3 \ _ 1 VSl / 7j)+ 12 
_ 1^ Vol J 47^+60 ) _ 1 \/l0Tfjt)2 19 I 
p'sVot 60 pH/p9'vToll8 + 20^ + ^r'' 
Vlll 153 I 1 VIY] (341 , 341 
2^^10Vll|36^ +140^+'r- pM/^ UTTI (1440?^" + 280^ + ^ 
_ j, V137 [ jj« 493 3349 | 
I2VT3 [162 1440 ^' 2520 ^^"^ r 
, , -8164,9658 1-2247,4487 
\/p ' p 
2-1908,9023 1-4907,1198 , 
7= T5 ^ ( + 3) To 
•8451,5425.^ -4183,3001 ,,^ 
j^Hp + 12) T, Y + 60) Ts 
p- 'Vp p' 
•3718,4890,,^ „ -1511,8579,,^, „ 
— (10jj=+ 171^j + 180)tc, '- (I75p2 + 1377^ + 1260) Ti, 
jf Vj; p 
-0569,8743 
•0201,0408 
(2387^= + 12276J9 + 10080) t,i 
(560// + 31059^-+ 120564^+ 90720) r,.- .... 
p. 
(ii) Laplacian Form of Expansion. 
This is an expansion with regard to the mode or maximum ordinate as origin. 
The mode of ?/ = '-^^ — r at x= (p — 1), so that it will be easier to deal with y 
in the form 
y^r(p- + i)' 
where p' = (p — 1). 
Let x = p' + ^, i.e. take the mode as origin. Then as before we require to find 
4>{-D) so that 
{p +^Ye-^v'+^^ r,.e-^^'ip' . 
r(/ + i) -^^-^^Vlg^ 
where D = ^ and z = "7=; . 
0,2 yp 
The introduction of Vp' in the denominator simplifies the integration a little. 
