ON THE GAUSSIAN CONSTANTS FOE THE YEAR 1829. 5 



orer it foUo'ws that, bj^ the knowledge of such anniial increase at two 

 moments separated from one another by a sufficiently hmg space of time, 

 we can calcnlate not only its value for any moment, but also that of the 

 corresponding total increase dnring any period. Therefore the materials we 

 possessed (as is to be shown hereafter) for computing the annual increments 

 of the constants for the year 1811 and for the year 1843-5, must suffice for 

 our reductions ; but before emplojdng them we had to consider the second of 

 the above-mentioned questions. An indubitable answer to it was, of course, 

 that we had to take into account and to determine the variation with time 

 for all those twenty-four constants the values of which were to be deter- 

 mined afterwards by the reduced observations. The solution of the problem 

 up to this highest degree of exactitude will at some future time be a beautiful 

 result of our present work, combined with a similar one for a later date ; but 

 had we undertaken it now, the preparatory task would not only have become 

 more extensive than the essential one, but would even have been impeded 

 by a most sensible want of means. Wc had therefore to content ourselves with 

 making our reductions for secular changes an approximation to reality, in 

 the same way as astronomers do when, in computing secular planetary 

 perturbations, they disregard the terms of less influence. So in this particular 

 case it was resolved to take into consideration only the changes of the first 

 two terms of the potential — that is to say, to ascertain for two epochs tho 

 annual increments of the first eight of the Gaussian constants. 



A. On the Equations for annual Increments of Constants durhir/ the year 1811. 



In order to ascertain the amount of the annual changes of the Gaussian 

 constants marked by 



ff''", r/\ h^\ ff\ f\ ^^'\ f\ and h'' 

 for our first epoch of 1811, we have founded the computation on : — ■ 



1. The increments of declination which appear as having happened 



from the year 1784 to the year 1840, from a comparison of the 

 maps of isogonic lines constructed for the said years by C. Hansteeu 

 and E. Sabine ; and 



2. The increments which inclination has undergone from 1780 to 1840 



and which appear as differences between the isoclinal maps of the 

 same authors. 



The increments of these two phenomena were taken by comparing the said 

 maps for forty-two points of intersection between the meridians of 



X = 0°, 60°, 120°, 180°, 240°, and 300° 



and the parallels of 



M=30°, 50°, 70°, 90°, 110°, 130- and 150°; 



and then, if t and t respectively designate the fifty-six years' increment of 

 declination and the sixty years' increment of inclination, and 



«!, a^' "^3) «4> l^i' "^G' «7 ^'^^ «8 



respectively the sought-for annual increments of the Gaussian constants 



