Frazer.] t)44 [April 6, 



Dr. Bowditch and Prof. Loomis were mainly instrumental in bringing 

 this subject prominently to the notice of scientific men. 



Prof. Loomis gives (Sillimau's Journ. of Sci. and Arts 1840) the annual 

 change of variation in 1840 as 2' for the Southern States, 4' for the Middle 

 and 6' for the New England States. 



The fessay is divided into : (a) Stations of reliable observations prior to 

 1740 ; (b) Ditto after that time ; ic) Results from comparatively recent ob- 

 servations ; (d) Establishment of formul* expressing the secular variation 

 within the limits indicated on the title page [i. e. the Atlantic and part of 

 the Gulf coast). 



Under (a), Providence, R. I., Hatboro', and Philadelphia, Penna., are se- 

 lected. Thirty declinations collected by Mr. M. B. Lockwood from actual 

 observations and recorded bearings of a number of objects, are considered. 

 The formula employed is 



D = do + y ( t — to) + z (t — to)- + u (t — to)^ -f &c. 

 Where y, z, u are unknown coefficients, and D = d where t = to. Put- 

 ting do = d/ + X where x is a small correction to the assumed value of 

 d, and omitting higher powers of the time, tiian the third 



_ D = d, + X + y It — t„) + z (t — to)- + u (t — to)^ 



Assuming for to the commencement of any year and for d/ the supposed 

 corresponding declination (expressed in degrees and decimals) then each 

 observed value for D at the time t furnishes the following conditional 

 equation : = d, — D + x + y (t — to) + z (t — t,,)' + u (t — t,,)^. 



From this by the application of least squares the normal equations and 

 coefficients of x, y, z, u, are obtained. 



" The above formula is capable of giving two maxima and two minima, 

 whereas the omission of the third power would give but a minimum. 

 And this as we know from observation took place about the commencement 

 of this century, to is assumed as 1830 ;* and d, = -|- 7.20." 



A table of thirty observations in Providence, R. I., is reduced in size by 

 substituting a table of sixteen means, which here follow : 



Substituting in the above equations we have : 



= — 2.40 -f X — 113 y + 12769 z — 1442897 u, 

 and similarly for the other fifteen equations. 



Multiplying each of the sixteen equations by the coefficient of the first 

 unknown quantity and adding them all up, we get the first of the normal 

 equations, and the same operation performed for each unknown quantity 

 will give the other normal equations. 



* In the discussion of subsequent years this date was changed to 1850. 



