120 Proceedings of the Royal Society of Edinburgh. [Sess. 
( 0 )*- 2 . 
145 x 4 
m=ll, 2185?/ = 337 — where x has values ranging from — 11 to + 11 
m = 12, 30015;/ = 4253-— xH-* 4 , 
4 4 
ra= 13, 930465^ = 121943- 
12635 
63 
4 4 
m =14, 445005« = 54251- 1^5 *2 + ?+, 
4 4 
m=15, 29667)/ = 3381 - — *2+-, 
4 4 
- 12 „ +12 
-13 „ +13 
-14„ +14 
-15 „ +15 
From these equations the formulae for k-1 and 2, m = ll . . . 15, given 
in Table I, were derived. 
§ 5. Incompatibility of Least-Square and Summation Formulae. 
Although the considerations adduced in § 4 made it tolerably certain 
that the least-square formulae (at any rate when the weights of the data 
are supposed equal) cannot be derived by summation processes, it seemed 
desirable to establish this fact directly, which may be done in the following 
way. Let E be, as usual, the symbolic operator defined by the equation 
Eu x = u x+1 , and write E = e 2i(f> so 
u —m + u —m + 1 + ■ • • + u — i + Uq + ?q + . . . + U m 
= | g-2 mi<t> + e -(2m-lb> _j_ . . + e -2 i<p + \ + e 2i<p _j_ _ . + e 2tni<p} UQ 
= (1 + 2 cos 20 + 2 cos 40 + . . . + 2 cos 2 m<}>)u 0 
sin (2 m + 1)0 
sin 0 0 
Hence if a graduation formula be written 
u o =P0 U 0+Pl( u l + U -l)+P2( U 2 + U -2)+ • • •» 
and if the right-hand side of this equation is capable of being expressed 
as the result of performing the operation \n~\ on a simpler expression, then 
when in this right-hand side we replace u n by e 2ni(l> u 0 we must obtain a 
function of 0, which vanishes when Sm n< ^ vanishes, i.e. when 
sm 0 
0 = 
7 T 277 
n ’ n 
that is 
Pq + cos 20 + 2p 2 cos 40+ . . . must vanish, for 0 = -, — . . . 
By the application of this test to each of the least-square formulae it 
was found that in no case can any of the least-square formulae be derived 
