J. D. Everett on Reducing Observations of Temperature. 25 
The general term in the expression is A, sin (nz+E,). t 
this fon assumed equal to P, cosnz+Q,sinnz. This assumption 
giv 
A, sin E,=P,, A, cosE,=Q, 
whence a= tan E,, A?—=P?24+Q?. 
The transformed series is 
y=A,+P, cosx+Q, sinz+ P, cos 22+Q, sin 22-4 ke. 
Let the given oe be denoted by 
Yo Y1 Y2 Yi1 
Then if the time 0 Ocrtglpond to the temiperature Y, the times, 
or values of z, corresponding to the 12 given values of y are 
Tespectively OP BO OOF te 4g Si 330°. 
et the sines of 0°, 30°, 60° and 90°, a denoted by the ab- 
breviations S,, S,, 8, ‘and S, . Then we have 
Yo =A, +P,8,40,8,4P,8, +95 ras ot PS; —Q,8, 
¥, =A,+P,S8,+Q,8,+P28,+Q25,+P,8,+Q,8,—P,8,+2,8, 
Ya =A,+P,S, +Q,8,—P,8,+Q,5,—P,8;—Q,8, - P,S,—Q,8, 
Ys =A,+P,8,+Q,8, —P,8, —Q,8, — PS, —Q,5,+P,8,—Q,S, 
¥, =A, —P,S,+Q,5,—P,8, —Q,S8,+P,8,+Q,8, —P,8,4+Q,8, 
Ys =A,—P,S,+Q,8 itP.S, —Q252.+P,5)+,5, —P,S, —Q,8, 
¥6 =A,—P,S,—Q,S,+P,8,—Q,5,—P,8;—Q,8, + P,8,—Q,8, 
Vz =A4,—P,S, —Q,5 :#+P.8,:+.,8, a P,8,—Q,8,—P,8,+,S8, 
Ys =A, —P,8,;—Q,5,—P,8, +Q,S8,+P,8,+Q,5, —P,8,—Q,S, 
V9 ag ae <° o—-Q,S8,-P, cee es ‘eae ate 
Yio= pe es —Q.5,— i+Q, 
¥s=A,+P,S,—Q,5,4P,8,—Q,8,—Pq8)—Qs8; —PaS,— 48s 
Pasa tid =a e 6 ae anne i ooeee asim’ 
which contain an 
larly, adding y, oye rs mie y, toY,, &¢., all the terms which 
contain P, Q,, P ill disappear, 
Let y,-y, =k, al ke X8,=4 
Y¥i—-y, =k, (k,-*,) X eee 
Yo-¥, =k, (ko—ky) KS, = ly 
¥s—¥, =k, ks XS, =/;5 
~¥ 55 k, And let 
Ye~wy, j= hy k 5, mm, 
a 
It will be found that the sum of J,, Z,, 7, a. I, is 6P,, and the 
sum of m,,m,,m, and m,is6Q,. | Hen P, andQ, are found 
Aw. Jour. Scr—Szcoxp Smries, VoL. XXXV, No. rhitdan, 1863, 
4 
