352 



Results of the Makerstoun Observations, 1844. 



TABLE XIX. — Number of differences in 1000 (without reference to sign), which occur between 

 the limits of successive minutes, for each Month, and for the Year 1844. 



Annual Variation of the Probable error of an Observation of Magnetic Declination. — From Table XIX. 

 1 have determined by a gi'aphic interpolation the probable error of a single observation from the monthly mean 

 for the corresponding hour, that is to say, I have determined approximately the error for which there are as 

 many greater as there are less than itself. These are : — 



Jan. Feb. March. April. Way. June. July. ^ug. Sept. Oct. Nov. Dec. 



C-QS 0'-94 l'-3.5 l'-20 l'-16 0'-78 1'04 l'-20 l'-36 l'-.58 l'-51 0'-90 



The probable error of a single observation from the monthly mean for the corresponding hour (without 

 I'cference to which hoiu') is a minimum at the solstices, and a maximum at the equinoxes : the principal mini- 

 mum occurs at the summer solstice, when the probable error is only 0'"78, or about half the value of the prin- 

 cipal maximum which occurs in October. In order to shew the difference of these values from those to be de- 

 duced by the aid of the calculus of probabilities, we may obtain the latter with sufficient accuracy by means of 

 the formula* 



Probable error = 0845 x mean of errors, 



which formula gives the following values ; — 



values which are from a fourth to a third more than the truth. 



Diurnal Variation of the Probable Error of an Observation of Magnetic Declination. — The following are 

 probable errors of a single observation for each hour (without reference to month) from the monthly means at 

 the corresponding hours, deduced from Table XXI. by a graphic interpolation. 



IXi.M. i'^ S'' 4l' .51' 61" 7l» 8'' 9l> 10>> llh 0>" Ih P.M. 2I> SI" i^ i^ 6l> ?!> 8l' 9'' lO"! Ill- 121' 



l'.12 O'Se I'-IO 0'-93 0'-96 0'-95 l'-04 I'-OO l'-17 I'-IO l'-17 1''04 I'-Ol l'"24 l'-24 0'-97 0'.79 l'-16 l'-.i4 i'-63 l'-41 1"33 l'-66 r-22 



Mcanofl j,.,,^ j,.^^ j-.^j i-.,,., j'.j^ j'.n y.y^ y.yi Q'-gg I'-n I'.JO I'-SS 

 two. ) 



From these values it appears that the probable eiTor of a single observation from the monthly mean, for 

 the corresponding horn- (without i-eference to the month to which the observation belongs), is a minimum about 

 .o*" A.M., and about 5*" p.m., being, in both cases, rather less than a minute; it is a principal maximum between 

 9'' and lO*" p.m., being nearly a minute and a-half (1''5), and a secondary maximum about noon, being about 

 1'13. 



On the whole, it appears that, if it were required to make a single observation which should he as near to 

 the monthly mean for the corresponding hour as possible, the observation should be made in June, and about 

 n^ A.M. or 5'' P.M. When a greater number of observations are obtained, more accurate values for each hour in 

 each month may be found. Upon examining the distribution of the errors in the months of May, June, and July, 

 5'' or 6'' P.M. is found to be the hour of the smallest probable error in each month ; the probable error at 5'' 



* Encke on " The Method of Least Squares." Taylor's Sciaitific Memoirs, vol. ii., p. 355. 



