Prof. W. A. Norton on Terrestrial Magnetism. 



11 



It remains to investigate a formula for the declination of the 

 needle. We have already seen that the magnetic needle is every- 

 where at right angles to the line of equal molecular magnetic in- 

 tensity traced upon the earth through its station ; which line we 

 have assumed to be the same as the isogeothermal line passing 

 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 annual 

 temperature of a place. This is 



T-=(^-T)(sin n <5.sin"(J / )-fT 



(4.) 



Fig. 8 



For the isogeothermal 



where t is the maximum equatorial temperature, r the minimum 

 temperature at each of the two poles of maximum cold, and 

 <?, d' the distances of the place 

 from the two cold poles. Let 

 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 . B A = £,and B A 7 = 8'. 



line, 

 since T is constant, dT=0. 

 Hence, if we differentiate 

 equation (4), and put the dif- 

 ferential equal to zero, we shall 

 have a relation between dS and 

 d8', the variations of 8 and S> [ n passing from the point B to its 

 consecutive point r on the isogeothermal line. Thus, putting 

 t — t=c, we have 



dT =c(n sin*" ■ 8 cos S sin* W + n sin"- ' 8' cos & siiTW). 

 Multiplying and dividing by sin" n+I 8 sin- n +"<? / , 



c( n cos 8 sin 8'dS -f- n cos 8' sin 8d8' ) 



Hence, 



sin-"*' 8 ®nr*+ l & 

 cos 8 sin 8'd8-\- cos 8 / sin 8d8 







a a dS 



sin 8 cos 8 



cos 8 sin 8 



(5.) 



If we drop the perpendiculars rs and rt upon BA' and BA pro- 

 duced, we have Bt=d8, and Bs=d<J'. Put Br = k 7 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 7 and BD, 

 the isogeothermal line will lie some where between these two 

 perpendiculars ; for it is only in this situation that in passing from 



