104 B. A. Gould—Algebraic Hapression of the 
or the true value of the daily mean, any one of these would sup- 
ply the place of an additional daily observation. And the simple 
graphical conditions, the diurnal curve has but one point of 
contrary flexure, affords yet additional facilities for attaining 
the desired end. 
The most important term of the formula, for various reasons, 
is the constant M, the daily mean temperature, which may be 
deduced with very considerable approximation from a rela- 
tively small number of observations. Thus in the mean of 
two values, corresponding to moments separated by an interval 
of half the day, all terms containing uneven multiples of / are 
eliminated, so that 
4 (T,+T en) =M + sin (2h+B)+dsin (444+D) +fsin (64 +F) 
+ ete., 
while from the half difference of the same values we obtain the 
total amount of the omitted terms, 
$ (T,—Tiyn)=a (sin h+ A) +e sin (3h+C) +esin (54+ E) + ete., 
a value which the, frequently very rapid, convergence of the 
series often renders extremely useful 
Thus from the mean of three equidistant values, all terms 
are eliminated in which the coefficient of A is not divisible by 
3; so that 
£ (Ty +T ist Tips) =M+esin (34+C) +/fsin (62 +F) + ete. 
Similarly the mean of four observations, at intervals of six 
hours, gives us 
three hours will differ from the daily mean only by the amount 
of the eighth term of the general formula; and that deduce 
really are, will soon be seen. 
