48 BULLETIN OF THK 



easternmost digression about 1191, showing at the time only 4° 

 west declination. Ever since this epoch the motion of the 

 north end has been westerlj'-, its present value being nearly 7|° 

 west. The greatest annual change, 5' nearly, has apparently 

 been passed. These stationary epochs are different for different 

 localities, the last one was noted earliest in Maine ; later in 

 Florida and Texas, and it has not yet been attained in California, 

 where easterly declination is still slowly on the increase. Thus, 

 the easterly stationary condition was reached at Portland in 1764, 

 at Boston in 1111, at Washington 1796, at Savannah 1809, at 

 New Orleans 1831, and at San Bias in Mexico in 1849. Ex- 

 cepting a certain region along our Pacific Coast, as indicated on 

 the chart above referred to, the effect of the secular change at 

 present is to increase the west declinations, or, what is the same 

 thing, to diminish the east declinations. The same seems to 

 take place in Alaska. 



This secular change is conveniently expressed by a circular or 

 harmonic function, viz : — 



B = g-f r sin (a m + c) + 7\ sin (a, m + c,)+ .... 

 when Z'r=the magnetic declination at anytime;! 

 m = the number of years (and fraction) from an adopted epoch 

 t^ = 1850, hence m = t — 1850-0. 

 a tti . . . . are factors depending on the adopted periods p pj . . . . 



., , 360° 360° , 

 so that o = — - aj = etc. 



P Pi 



r i\ . . . are parameters and c c^ . . . epochal constants of the 



several periodic terms. 



g = a constant representing the mean or normal direction of the 



needle about which the secular motion takes place. 



Thus, for each place for which we have a sufficient number of 

 observed declinations we have to determine four unknown quan- 

 tities, viz., 5, r, a and c for the establishment of the first or prin- 

 cipal term and three for each following term. This is done by 

 application of the method of least squares, each observation fur- 

 nishing a conditional equation of the form 



o = 5j — D -\- X + sin a. m. y -j- cos a in z -{- 



supposing a has been suitably assumed and where 6 = 6j-f x 

 yz=r cos c and z = r sin c. The process must be repeated for a 

 value a -{- d a and so on until the sum of the squares of the differ- 

 ences of the observed and computed values equals a minimum. 

 The second periodic term may best be established by Cauchy's 

 method of interpolation. 



The annual change vis found by 



■i;= 60 sin 1° [r o cos (a m + c) -f 0\ a, cos (a^ 771 + Cj) + . .] 



expressed in minutes, and maxima and minima are found by putting 

 the expression within the brackets equal to zero, from which 

 equation m can be found. The apparent probable error of an 



