1919 - 20 .] On a Class of Graduation Formulae. 
121 
by the summatory process. The difference between the two classes of 
formulae is therefore fundamental. 
In applying the <f> method of testing whether a least-square formula of graduation 
can be represented by a summation formula, we may note that the terms of the least- 
square formula may readily he summed with respect to <f> and thus presented in 
a compact form which is readily tested : for example, the least-square formula of 
graduation corresponding to Jc= 1, m= 10 may be written 
1 ( 221 sin 21 ft lQg cos 21 (p cos <p 5 sin 21 <p cos 2 </>\ ^ 
U ° 3059V sin (f> 2 sin 2 (p + 2 sin 3 (f> ) 0 
where e 2m '^w 0 represents u n . It is only the work of a moment to show that this 
does not vanish when <& = - , where n is a given whole number. On the other 
n 
hand, in considering Spencer’s formula and other summation formulae, it was found 
that the series 
, 0 2?r n 4tT „ COS 67T 
Po + 2ft COS — + 2p 2 cos— + 2jp 3 + 
n n n 
took the forms 
2lP 
l + 2( cos y + cos 
)i . . . *,[ 
. . a 2 tt 4tt 
1 + 2 COS r j-rr + COS -r+ • 
2n+\ 2w + 1 
. 2mr 
+ cos 
2n + l 
)] 
^i[l + 2(cos|)] : A,[ 
Ty r 1 L o( 2-7T 47 r 6tt 
1 + 2^C0S y + COS y + COS y 
where the It’s are numerical quantities, when it had been obtained by the operations 
[3], [5], [7], . . . [272 -hi] respectively on simpler expressions. Each of these 
expressions vanish as the part in the round brackets in every case equals — J. 
A more straightforward test* is one obtained from the application of the 
remainder theorem ; e.g. Spencer’s formula would he looked upon as corresponding 
to the algebraic expression 
— 1 - 3x - 5x 2 -5x 3 -2x i + 6x 5 + 18x 6 + S3x 7 -j-47x 8 + 57x 9 + 60x 10 + 57x 11 + 47x 12 + 33x 13 
+ 18a4 4 + 6a? 1 5 — 2x u - 5a? 17 — 5a? 18 - 3a? 19 - x 20 . 
If Spencer’s formula can be derived by performing the operation [5] on a simpler 
expression, then this algebraic analogue of Spencer’s formula must be divisible by 
1 ' ~ 
1 +x + x 2 - s roc 2 -\-x i , i.e. divisible by 
1 — x 
For the test substitute x 5 for 1, arrange 
the expression so obtained in columns of x°, x, x 2 , x 3 , and x 4 , and add the members 
in each column, e.g . : — 
x°. 
x. 
X 2 . 
z 3 . 
a? 4 . 
- 1 
- 3 
- 5 
- 5 
- 2 
+ 6 
+ 18 
+ 33 
+ 47 
+ 57 
+ 60 
+ 57 
+ 47 
+ 33 
+ 18 
+ 6 
- 2 
- 5 
- 5 
- 3 
- 1 
+ 70 
+ 70 
+ 70 
+ 70 
+ 70 
* See also Trans. Act. Soc. of America, vol. xix, pp. 302-3. 
