of Temperature for unequal length of Months. 373 
and the mean of them all, or the annual mean, is 48-868. In the 
manner just explained I have found that the means for mean 
Months are ( 
36°58 46°52 ~~ “6180 49°84 
38°62 53°12 6199 43°20 
41°47 59°13 56°39 39°28 
and the annual mean is 48-912. The mean months make the an- 
nual mean greater by ‘049 than the calendar months do. We 
evidently cannot infer that the mean of the monthly means will 
Ss equal to the annual mean unless the months are all of equal 
ength, 
To find the equation of the curve for Greenwich I have fol- 
lowed the method given by Prof. Everett, determining first the 
equation which expresses the series of monthly means, and then 
multiplying the “amplitudes” A,, A,, Aj, &., by the ratios 
a ° °o 20 a “4 
eh MS i: &e., respectively, and substituting <—15° 
sin 15°’ sin 30° sin 45° 
instead of 2, so as to place the origin of codrdinates at the be- 
ginning of January. Finally, I have transformed the equation 
rom 
y=A,-+A, sin(z+-E,)+-A, sin(22-+-E,)+-A, sin(3z++-E,)+ &e., 
into the form 
=A, -bA, sin(e—e, )+-A, sin 2(a—e,)-A, sin3(z—e,)+ ke. 
This is readily done, because when E, is greater than 180° we 
shall have 
sin(nz--E, )=sin n[z—}(360°—E,) ], 
and when E, is less than 180° we shall have 
sin(nz-LE,)== —sin n[a—3(180°—E,)]. 
The advantage of this transformation is that the equation will 
Show the “ date of phase” ata glance, the are e, being the meas- 
ure of the time from the beginning of the year to the date when 
the term A ,sin(x—e,) first becomes zero. In general, the are ¢, 
expresses the time between the beginning of the year and the first 
zero-point of the term A, sin x(x—e,). Furthermore, the posi- 
tive or negative sign prefixed to any term will show whether 
the value of that term is increasing or diminishing at the point 
where it first becomes zero; in other words, to borrow a phrase 
from Astronomy, the sign will show whether the first node is an 
ascending or a descending one. ; : 
The means for the calendar months give the following as the 
final equation of daily temperatures at Greenwich. 
= 48°86 ; 
+-12°59 sin(x— 112° 38’/)—°85 sin 2(r— 76° 1/)-'17 sin3(z— 23° 40’) 
+28 sin 4(r— 40° 42’) — -26 sin 5(7— 0° 39’) —-23 sin 62. 
