Two DEGREES OF THE NORTH POLE. 5 1 
will give rise to the following eight equations of condition of which 
there are four repetitions, u standing in turn for d^i, dr^, dA', and duj. 
«1 + "2 + "3 + «4 + «5 + «6 + "7 + «8 = O , 
Ui — Uio + Vi) = O , 
«1 + 1(2 — Un + «9 =o , 
Ul -\- U2 + Us — Ui2 + W9 = O , 
«1 +M2 + W3 + «4 — Z<i3 + i<9 =0, (3) 
«1 4- «2 + «3 -{- ?'4 + «5 — Wi4 + ^l9 = O , 
Ui + 112 + W3 + t(4 + «5 + Z<6 — «15 + ?'9 = O , 
Wl + «2 + «3 + «4 + «5 + ^6 + W7 — M16 + U9 = O . 
The second terms of these equations will not actually be equal to 
zero, but there will exist small discrepancies Wi . . . ws, and these 
must be distributed among the various quantities «i ... Uk, in such 
a way as to cause the final dififerences to become as small as possible. 
This problem is solved by the method of correlatives (Merriman, 
Least Squares, pp. 59-64). Let ki . . . kshe the multipliers or cor- 
relatives of the equations of condition. We shall then have the fol- 
lowing eight normal equations which are to be solved by the ordinary 
processes of elimination. 
8fei + k2-{- 2^3 + 3^4 + 4^5 + 5^6 + 6^7 -\- jks + -Wi = O , 
3^2 + 2^3 + 2^4 + 2^5 -f 2^6 + 2^7 -f 2^8 4" ^2 = O , 
4^3 + Sh 4- 3^5 -r 3^6 + 3^7 + 3^8 4- "2^3 = O , 
5^4 + 4^5 4- 4^6 4- 4^7 + 4^8 + W4 = o , 
6^5 + 5^6 4- sky + 5^8 4- '^5 = o , (4) 
7^6 + 6^7 + 6^8 + •U'6 = O , 
8^7 4- 7^8 + W7 = o , 
9^8 4- ■"^8 = o . 
The values ki . . . kg, when substituted in the following formulas, 
will give the corrections to the various quantities ui . . . wig. 
dui = fei 4- ^2 4- ^3 4- ^4 -f ^5 4- ^6 4 ^7 4- ^8 , 
du2 = ki -{- ki -{- k^ -\- ks -\- ke -{- kj + kg , 
dUi = ^1 + ^4 4- ^5 + ^6 4- ^7 + ^8 , 
du^ = ki + ks + kf, -\- Ut -{- k% , 
dus = fei 4- ^6 4- ^7 + ^8 , 
dU6 = ki -\- kj -\- ks , 
