Lovering and Bond on Magnetic Observations at Cambridge. 67 
netic Associations, Gauss calculates on the principle of least squares, 
which allows more places on the earth to be represented than there 
are unknown quantities, the values of his coefficients. After pass- 
ing the formulz through several new forms, the chief object of which 
was to make them more simple and to facilitate the application, he 
brings the three components of the function into the following 
shape. 
X, Y and Z are the three coordinates of the magnetic force ex- 
erted upon a given point of the earth whose longitude is reckoned 
east from Greenwich. The auxiliary angles 4’, 4", &c. B', BY, &e. 
C', C", &c. depend upon the latitude. 
X = a°+a'cos.(4-')+-a" cos.(24--4")-+-a" cos. (34-+-A")-+-a” cos.(444+ A")... 
Y = b'cos.(4-++-B')-+-b" cos.(214-B")-++b" cos.(3i-+.B")+b* cos.(41-+-B")... 
Z= c°+c' cos.(a+C’)-+-c" cos.(21-+ C")4-c™ cos.(34-+ O")-+c" cos.(44+ C")... 
Professor Peirce has calculated the value of X, Y and Z by these 
formule for the Cambridge Observatory, whose longitude is 71° 7.5 
W. and whose latitude is 42° 22’ N. Here 2 — 288° 52’.5. The 
equations give these values for the coefficients and auxiliary angles. 
a + 645.9 6° —0 c= + 1300 
log. ai = 2.29185 log. bi = 2.23230 log. cl — 2.36763 
log. a! = 1.74467 log. bii = 2.04629 log. cli = 2.19893 
log. a'" = 1.36560 log. bii — 1.59317 log. ciii — 1.62487 
log. a'Y = 0.75277 log. b'v = 0.92425 log. civ — 0.88968 
Ai = 245° 55/ Bi = 358° 17! ci = 90° 34! 
Aii = 278 42 Bi — 56 55 ci — 157 13 
Aili = 243 31 Biii = 322 25 cii—= 46 19 
Aiv = 142 26 Biv = 232° 26 Civ == 322 26 
then we have: 
4+A4i = 534° 471.5 a+ Bi 642° 91.5 A+C? — 379° 26/.5 
Q14+.4ii —= 856 27 2+Bi — 634 30 24+Ci — 734 58 
3i4f4ii = 1110 8.5 | 324+Bii—1188 40 314+Cii— 912 56.5 
4a+ Aiv — 1297 56 41+ Bit — 1387 66 424+C — 1477 56 
and consequently : 
+ 645. 9 =a» 
log. a + log. cos. ( 2+ Ai ) = 2.29006 = — 195.01 = ai cos.( 2+ 4) 
log. a'' + log. cos. (22 + A) = 1.60487 = — 40.26= aii cos, (22 + A’) 
log. ai! + log. cos. (34 + Ai) = 1.30251 = + 20.07 = ati cos. (32 + Ai) 
log. a'vY + log. cos. (42 + 4i*) = .64970—=— 4.46 = a’ cos. (42 + 4") 
X= 426.20 
