14 AMPLITUDE OF THE SOLAR-DIURNAL VARIATION 



the signs changing at the time of the equinoxes. To ascertain how nearly this is 

 true, and in order to obtain a more precise expression, the means of the two columns 

 (after changing the signs in the second) for each month respectively, were put into 

 an analytical form, using Bessel's well-known formula for periodic functions 



A a = +l'.78 sin (e + 90) + 0'.32 sin (29 + 180) ; 

 or, A a = -f 1'.78 cos e 0'.32 sin 29 ; 

 the angle 6 counting from January 1st. 



The maximum values will occur on the first of January and the first of July; 

 and the transition from a positive to a negative value, and the reverse, will take 

 place on the first of April and the first of October, the equation 1.78 cos 6 = 0.32 

 sin 2 S, being only satisfied for = 90 and 270. That the angles <7, and C a 

 should be exactly 90 and 180 is remarkable. The monthly values are satisfied 

 as follows : 



Middle of By observation. By calculation. 



January + 1'.50 + 1'.56 



February +1.22 +0.94 



March +0.47 +0.30 



April 0.46 0.30 



May . . . . . . . . 1.16 0.94 



June 1.40 1.56 



July 1.59 1.56 



August 2.00 0.94 



September 1.03 0.30 



October +1.28 +0.30 



November +1.41 +0.94 



December +1.76 +1.56 



The regular progression of the monthly values is a feature of the annual varia- 

 tion deserving particular notice. There is no sudden transition from the positive 

 to the negative side, or vice versd, at or near the time of the equinoxes (certainly 

 not at the vernal equinox) ; on the contrary, the annual variation seems to be 

 regular in its progressive changes. The method here pursued is entirely different 

 from that employed by General Sabine for the same end, but the results are, never- 

 theless, in close accordance. He remarks (in the British Association report above 

 cited): "When a mean is taken corresponding to the 10th or llth day after the 

 equinox, the transition from the character of the preceding six months has already 

 commenced and advanced very far towards its completion, and, by the middle of 

 October, is quite complete; apparently, the progress of the change is somewhat 

 more tardy in the March than in the September equinox." From the above an- 

 alysis, we have found that the transition took place ten days after either equinox, 

 and also that the turning points occur ten days after the solstices. 



For the more precise determination of the law of the phenomenon, and in order to 

 render the results of similar investigations comparable with one another, the regular 

 solar-diurnal variation is now to be expressed as a function of the time. The pre- 

 ceding tabular values, given in minutes of arc, when treated as required by Bessel's 1 

 periodic function, furnish the following expressions for each month of the year: 



1 For another development of the formula, see Rev. Dr. H. Lloyd, " On the Mean Results of Obser- 

 vations," Transactions Royal Irish Academy, 1848, Vol. XXII, Part I. Dublin, 1849. 



