180 
On Expansions in Tetraclioric Functions 
TABLE III. 
4r/rn. dx, z = - 2-6846788 *. 
Jo r(49) 
Tetraclioric 
TeriBs in 
Value of Series 
s 
Functions Tg 
Series - Os'rg 
up to term Tj 
Q 
_i_ -00*^(190^ 
1 •00'^fi9Qf! 

•On'^ft9QR 
yjyjoy) ^o\} 
~\- -01 08'iQ7C) 
— \'\J ±K}yj i O 
•0090R91 
2 
— -09081 ^7"^ 
1 -000^074 
-009fif;Q'S 
4- •09759n8Q 
— '00'^44fil 
— •00077fifi 
4 
4- -OOl ^440 
•0007R7'S 
5 
— '000*^07 *^ 
yjyjyjoyj t o 
•0004R01 
Q 
n-049'^fi'^m 
-4- ■00'^'S70fi^ 
_ •0009'^fiO 
-0009941 
7 
— '01 4(^0*^70 
1 •OOOfSO'S'^ 
^ yjyjyjoyjoo 
-00079Qfi 
-4- •00Q*^Q'S04 
yjyjtjtjtjoyj^ 
— -00091 50 
— yjyjyj^Loyj 
-00051 44 
9 
— 'OOOOQl 
•0004.99Q 
10 
0'09-LS8ftS4 
— '01 1 01 974 
1 -0009741 
-OOORQRQ 
0-091 '^fi9ftQ 
4- '005700^5 
— '0001 9*^7 
— yjyjyj L I 
•000'i7'^9 
12 
A- ■OO^OQ7'^7 
— -0001 1 05 
— yjyjyj L 1 ij'j 
•0004fi97 
13 
0-09979779 
— '0071 *^41 Q 
1 -0001 R91 
-000R9d.Q 
14 
0-02256727 
+ -00058475 
- -0000132 
-0006117 
15 
0-02351439 
+ -00599464 
- -0001410 
•0004707 
16 
0-02546065 
- -00455186 
+ -0001158 
•0005865 
17 
0-02739094 
- -00248832 
+ -0000682 
•0006547 
18 
0-02996700 
+ -00573797 
- -0001720 
•0004827 
19 
0-03360181 
- -00124666 
+ -0000419 
•0005246 
20 
0-03803862 
- -00454994 
+ -0001731 
•0006977 
■21 
0-04355963 
+ -00382135 
- -0001665 
•0005312 
22 
0-05068120 
+ -00204641 
- -0001037 
•0004275 
23 
0-05968962 
- -00471305 
+ -0002813 
•0007089 
24 
0-07107603 
+ -00066657 
- -0000474 
•0006615 
25 
0-08566837 
+ -00406751 
- -0003485 
•0003130 
26 
0-10443752 
- -00276906 
+ -0002892 
•0006022 
27 
0-12866399 
- -00240728 
+ -0003097 
•0009119 
28 
0-16018283 
+ -00383980 
- -0006151 
•0002969 
29 
0-20147298 
+ -00036667 
- -0000739 
•0002230 
30 
0-25589543 
- -00382481 
+ -0009788 
•0012017 
True value •0005850. 
near to the ' true value ' line, the approximation is not really a good one. The 
important question for us is : To how many decimal places does the series give the 
result correct ? On going through the tables it will be found that there is no value 
of the series up to the ^th term giving the result correct to more than three or four 
places. We now come to the real trouble. Suppose a frequency function is expanded 
in tetrachoric series, how are we to know at what term to stop so as to obtain the 
most accurate result ? If the value of an integral is required, the true value is 
wanted. In our work we chose integrals of which the value was already known. 
From Figs. 1 — 4 it is easily seen that we have as good an approximation at the 
