Lovering and Bond on Magnetic Observations at Cambridge. 45 
D,, = the mean of the observations taken at the time h of each day. 
Then if S$‘ denote the sum of all the terms which correspond to 
the different values of h, we have 
1. m 4 — S' D, 
m representing the number of intervals on each day. 
2.mC, cos.2anc¢,—2 S' D, sin. 2 anh. 
3. m C, sin. 2 anc, =—2 8S’ D, cos.2 anh. 
There is no known periodic function which does not admit of de- 
velopement according to the sines and cosines of the time and its 
integral multiples and in the absence of positive evidence the same 
thing may be assumed in regard to that under present considera- 
tion. The constant 4, being equal to *“* is the mean of all the 
m9 
partial results obtained from observation for the several intervals into 
which the day is distributed for this purpose. By substituting dif- 
ferent values for » we obtain an indefinite number of terms out of 
the general one C, sin. 2 77 (é+c,). It appears, however, from 
the calculation that the series rapidly converges so that the first four 
or five terms are sufficient to give the declination within a degree 
of exactness corresponding to the accuracy of the observations them- 
selves. Dividing the 2d equation by the 3d, we have the value of 
the tang. 27nc,; and multiplying equation 2d by cos. 277 c,, and 
equation 3d by sin. 2 zn c,, and adding them together we readily 
find the value of C, Thus, if the numbers 1, 2, 3, be succes- 
sively taken for n, we shall have the following equation for finding 
the approximate declination, or the empirical magnetic curve : 
D—A+C, sin. 2 a (t+ ¢,)+ C, sin. 41 (+ c,) + C, sin. 6 a (t+ co). 
The empirical thermometric curve is calculated on the same princi- 
ple by this formula : 
T=B + D, sin. 22 (t+d,) + D, sin. 42 (t+ d,) + D, sin. 6 2 (t + d,). 
