So7ne Besults from the Periodic Formulce. 119 



And therefore the expression on the right-hand side is very nearly 

 the summation of the sine series evaluated for any month at Kim- 

 berley. We shall obtain from this equation the following set of 

 theoretical values corresponding to the middle day of each month : — 



o o 



Jan 92-3 July 65-8 



Feb 89-6 Aug 71-8 



Mar 83-7 Sept 80-0 



Apr 75-5 Oct 87-2 



May 67-3 Nov 91-3 



June 63-3 Dec 92*7 



And if we compute the numerical coefficients and angles in the 

 periodic formulae for these we get, to the fourth periodic term — 



Ih = + 13-5377 



^i = 



: - 5-8467 



]J,= - 1-1750 



^2 = 



+ 1-5155 



p^ = _ -2000 



^3 = 



+ -0167 



2), = + -1583 



1, = 



: + -0433 



a = 80° + 14-747 sin (113° 22' + 7;i30) 

 + 1-918 sin (322° 13' + mm) 

 + -201 sin (274° 46' + vm) 

 + -164 sin ( 74° 42' + wl20) (2) 



where evidently only the coefficient of the last term differs by a 

 material percentage from that given in Table 2. 



Denoting the amplitudes and phase-times of this particular 

 formula by a-,, a^, . . . A^, A2, ... as before, and supposing the 

 successive differences between these theoretical values and the 

 values from observation, given in Table 2, to be ft^, [j^, • - • we 

 find for the normal maximum temperature on any assigned day, 

 very nearly — 



a = AS^ cos Z -f B - 2 ja, cos (a,-^+ mSO) sin ^^ 



+ a, cos (^A, - ^-^' + m60^ sin ^' + " • • } (^) 



in which twice the quantity within the ceratic brackets is the 

 correction to be applied to either of the standard forms — 



AS^ cos Z 4- B 



or — i? + «! sin (A^ + iu30) + • • • 



in order to obtain the observed series — 



p + a, sin (A, + /3, + w30) + (4) 



