184 3 REPORT—1857. 
EXPLANATION OF PiATE II. 
Fig. | represents the mean daily curve for each month, as deduced from 
Table XIV. From among the almost straight lines representing the winter 
months, March is observed to rise in a bold curve, showing the rapidly in- 
creasing power of the solar rays, which had hardly produced any effect in 
shaded places during February. April exceeds March by 2 degrees in the 
height of its curve; but that of each summer month in succession becomes 
flatter in consequence of the summer warmth being attended by more cloudy 
weather, and a lessened fall of temperature during the nights. September 
presents a curve of great flatness, the difference of temperature between noon 
and midnight scarcely exceeding 1 degree. The almost continued foggy state 
of the weather then prevents the sun effecting much rise in the day, and the 
great extent of sea, river, and lake surface, still unfrozen during the earlier 
part of the month, prevents any great fall of temperature during the night. 
With the exception of a slight rise in the October curve, the whole six repre- 
senting autumn and winter are remarkably flat. 
Fig. 2 shows the mean daily curves of the four seasons, deduced from 
Table XVI.; also the curve of the winter half of the year, or the autumn and 
winter combined, the curve of the summer half, which is spring and summer 
combined, and the curve of the whole year, each being the mean of two years. 
The flat curve of autumn has nearly the same form as September, and in 
position occupies the place of the mean temperature of the year. The winter 
line nearly coincides both in form and position with January, and is very 
flat. Spring has nearly the form of March and the position of April; and 
the summer curve, either in shape or position, does not differ much from 
July. With the exception of autumn, the curve of each season bears a 
striking resemblance to that of its middle month. 
Fig. 3 indicates the curve of mean daily temperature shown by a black- 
ened thermometer exposed to the sun. The black line represents the month 
of June, and the dotted one three weeks before and after the 21st of June. In 
both, the extremes seem to be very near noon and midnight, and the bold 
character of the curve is very striking. 
On the Algebraic Couple; and on the Equivalents of Indeterminate 
Expressions. By Cuarues JAMES HarGREAVE, LL.D., F.R.S. 
In a paper entitled “ Analytical Researches concerning Numbers,” which 
was published in the Philosophical Magazine some years ago (vol. xxxv. 
p- 36), I had occasion to avail myself of a principle, which, though it has not 
yet taken rank amongst recognized forms of mathematical reasoning, appears 
to be capable of extensive application and calculated to lead to true and use- 
ful results. This principle may be expressed by stating simply, that the ana- 
lytical equivalent of an indeterminate expression is the arithmetical mean of 
all its possible values. The accuracy of the results to which I was conducted 
by the application of this principle in the paper above referred to, led me to 
the conclusion, that the principle might probably to some extent be intro- 
duced into mathematical science, without departing from or unduly extend- 
ing doctrines heretofore admitted. Since that period, I have not had the 
opportunity of pursuing this interesting subject; but I find that it is one 
