Prof. W. A. Norton on Terrestrial Magnetism. 11 
through the same point. We have therefore only to seek for a 
formula which shall make known the direction of the isogeo- 
thermal line at a given place and place the needle at right angles 
to this line of direction. Such a formula may be derived from 
Brewster’s formula for the determination of the mean annua 
temperature of a place. is 1 
T =(¢—1)(sin"d. sin"d’)+1 : ee 
where ¢ is the maximum equatorial temperature, t the minimum 
temperature at each of the two poles of maximum cold, and 
0, 0 the distances of the place Fie. 8 
from the two cold poles. Let i 
C, fig. 8, represent the north 
pole of the earth, A and A’ the 
two poles of greatest cold, B 
a given place, BL the direc- 
tion of the isogeothermal line 
through B. BA=d,and BA’=0’. 
For the isogeothermal _ line, 
since 'T’ is constant, dT =0. 
Hence, if we differentiate 
equation (4), and put the dif- 
ferential equal to zero, we shall 
ave a relation between dd and 
ds’, the variations of 6 and 9 in passing from the point B to its 
consecutive point r on the isogeothermal line. ‘Thus, putting 
—t=c, we have 
dT =e(nsin"~'d cos 6 sin"d’dd +n sin"~ "0" cos 0” sin"ddd"). 
Multiplying and dividing by sin-*t' 3 sin-"+ 10’, 
av’ e(n cos 5 sin 0’d5 +n cos & sin 6dd’) 
a gnc" t !. d. gin PAR 5 
Hence, cos 6 sin 5’dd + cos 0 sin ddd’=0 
ds sin 0 cos 0’ 
pedo gi atc ne tne ge 5. 
And, = —-<ydan¥ (5.) 
If we drop the perpendiculars rs and rt upon BA’ and BA pro- 
ced, we have Bt=d9, and Bs=d%. Put Br=k, angle rBt=a, 
and angle rBs=a‘. If in the angle A’BD we conceive two arcs 
to be drawn through B respectively perpendicular to BA’ and BD, 
the isogeothermal line will lie some where between these two 
Perpendiculars ; for it is only in this situation that in passing from 
