TWO DEGREES OF THE NORTH POLE. 51 
will give rise to the following eight equations of condition of which 
there are four repetitions, u standing in turn for d=, dy, dA’, and dw. 
Uy + U2 4- U3 + U4 + U5 + U6 + U7 + Us i), 
Uy — U9 + Ug =O, 
Uy + U2 — U1 + Ug — 0 
U, + U2 + U3 — U12 + Uo =o, 
Uy -+ U2 + Us + Us — 13 + Uy =O, (3) 
Uy + U2 + Uz -+ U4 + Us — W114 + Ug EM Ae 
Uy + U2 + U3 + U4 + Us + U6 — U5 +- UO =O, 
Uy + U2 + U3 + U4 + U5 + U6 + U7 — U16 + U9 iW ee 
The second terms of these equations will not actually be equal to 
zero, but there will exist small discrepancies w, ... ws, and these 
must be distributed among the various quantities u; ... w#15 in such 
a way as to cause the final differences to become as small as possible. 
This problem is solved by the method of correlatives (Merriman, 
Least Squares, pp. 59-64). Let k; ... ks be 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. 
8k, + ko + 2k3 + 3k4 + 4hs + 5k6 + 6k7 + 7kg + Wi =O, 
3k2 + 2k3 + 2k4 + 2ks + 2k6 + 2k7 + 2kg + w2=0, 
4k3 + 3k4 + 3ks + 3h + 3h7 + 3ks8 + w3 =0, 
5k4 + 4hs + 4ke + 4k7 + 4kg + We =0, 
6ks + 5k6 + 5k7 + 5k8+Ws=0, (4) 
7ke + 6k7 + 6kg + We =O, 
8k7 + 7kg + w7 =0, 
okg + Wg=O. 
The values k; ... ks, when substituted in the following formulas, 
will give the corrections to the various quantities uw, ... U16. 
du, —ki+k2+k3+hkatks+khe+k7+ks, 
duz =ki+k3+kg+hs+kho+h7+ks, 
du3 =ki+ky+ks+hke+hk7+hks, 
dug =ky+ks+khko +hk7+khks, 
dus =ki+hko+k7+ks, 
dus =ki +k7+ks, 
