180 



MR THOMAS HEATH ON 



similar determinations by means of the old set, we may reasonably conclude that the 

 corrections have not been appreciably erroneous, and that the new thermometers may 

 be confidently accepted as fairly competent to take the place of the old ones, allowance 

 being made for the difference of the depths of the two sets. 



Equations to the Curves. 



We may put the equation to the curve in the form 



T = T + a 1 sin(27r/+?- 1 ) + a 2 sin(47r/ + ?- 2 ) + &c. 

 where T is the mean temperature of the year, a x and a 2 ^ ne half- amplitude of the 

 annual and semi-annual terms respectively, f the fraction of the year, and 7\ and r 2 the 

 retardation of phase of the same two terms. If we put c^ = a x sin r v and /5 X = <x x cos r 1? 



Ct 



then s/( a l + PI) = a i an d nr = ta,nr 1 . Substituting these values of a x and r v and 

 similar values of a 2 and r 2 , we reduce the equation to the form 



T = T + (a 1 COs27r/ + /3 1 sin27r/) + (a 2 cos47r/'+^ 2 siii47r/) + &c. 



Giving to f the successive values 0, ■£%, T 2 2, \^, we form the following 



twelve equations : — 



T-T = +a 1 +a 2 



= + a x cos 30° + ft sin 30° + a 2 cos 60° + ft sin 60° 



= + a x cos 60° + ft sin 60° - a 2 cos 60° + ft sin 60° 



= + ft -a 2 



= - a x cos 60° + ft sin 60° - a 2 cos 60° - (3 2 sin 60° 



= - dj cos 30° + ^ x sin 30° + a 2 cos 60° - /3 2 sin 60° 



= -a x +a 2 



= - a x cos 30° - ft} sin 30° + a 2 cos 60° + /3 2 sin 60° 



= - a x cos 60° - (3 1 sin 60° - a 2 cos 60° + /3. 2 sin 60° 



= -ft -a 2 



= + a x cos 60° - (3 1 sin 60° - a 2 cos 60° - /?., sin 60° 



= + a 1 cos 30° - ft sin 30° + a 2 cos 60° - ft sin 60° 



The eight sets of values of T — T are contained in the two tables on pp. 177 and 178 

 for the old and new thermometers respectively. In working out the equations, Prof. 

 Everett's method of elimination has been followed, and the results are shown in the 

 following tables : — 







Old. 







New. 





*1 



h 



3 



h 



h 



h 



h 



h 



a l 



fit 



+ 0-6265 

 + 0-0112 

 -0-0063 

 -00193 



+ 0-5060 

 - 2-2480 

 -00938 

 + 0-0757 



-2-3535 

 -4-1797 

 -0-0022 

 + 0-3996 



- 5-4648 



- 4-5537 

 + 0-3077 

 + 0-6347 



+ 0-8458 

 -0-4003 

 -0-0288 

 -0-0298 



+ 00135 

 -2-6992 

 -0-0682 

 + 01 140 



-3-5798 

 -4-2038 

 + 0-1947 

 + 0-4140 



-5-5810 

 -4-1902 

 + 0-4128 

 + 0-5890 



