48 RESEARCHES ON II. 



log. / {,%, S0°) .... =9.3775010 



— Log. f {c, B0°) ==8.8445407 



%. O = 0.5329603 



*= 7^1? = '■*"'''■ 



Ohservation. — Fi'om an accurate inspection of this calculation, it is evident that 

 even the last figures of numbers and logarithms have a very great influence in the 

 result ; we had made some of these calculations hmiting the numbers only to five 

 figures, but the results were found sensibly dijfferent from those obtained with six 

 figures, and the difference sometimes arose to two units in the first decimal note. 

 This is principally due to the value of the modulus c and its complement b, 

 which vary very rapidly as they approach to 90°, and their variation introduces a 

 still greater one in the elliptic functions. The function F chiefly varies very rapidly 

 near that limit, and the difference is already sensible at the third logarithmic figure, 

 even for arcs differing less than a minute of a degree after 75°, and in general 

 the difference of a minute in the arc d induces a variation in the 4th decimal figure 

 in the logarithms of both functions F^ and E'-, through the whole quadrant. Now 

 as the sine varies very slowly in the higher degrees, a little variation both in 7i^ or 

 h is sufiicient to introduce a great difference in the arc. To this source of errors, 

 we must add that which depends upon the tables of elliptical functions. These 

 functions are only calculated for every tenth of a degree or 6 minutes, and very 

 often we felt the necessity of having them calculated for smaller divisions. To 

 compensate for this want of smaller fractious, we have taken the arithmetical parts 

 of the first differences, as they are given in Legendre's tables, which are sufiicieutly 

 accurate when the value of their logarithms is limited to seven decimal places ; 

 this, however, we confess is not quite exact, since it induces a small error in the 

 last decimal places. It would have been, however, too tedious and troublesome to 

 calculate by the formula of interpolation their proportional parts, and besides, this 

 would have been a useless labor, since it is impossible to avoid errors of that order 

 in a long logarithmic calculation. Consequently, we do not present our results as 

 accurately exact even to the last figures, but we think that they will be so as far 

 as the fourth. 



To obtain greater exactitude, it is necessary to use larger tables having more 

 than seven decimal places ; but this is not required, until the degree of accuracy 

 in the experiments is carried a good deal further than it was in our power to do. 



The calculation for the central position of the needle is a good deal shorter, since 

 a =: and 7i^ = Rm only. 



In this manner we have calculated three series of experiments made with the 

 circle on different points of its vertical diameter, and the result is given in the 

 following table. The numbers represent the ratio of intensity of forces in different 

 points of the diameter, that in the centre being taken as unity. 



In the fiirst horizontal line of the table is the number of the experimental series, 

 as taken from the tables (§ preceding) ; in the second is the absolute value of the 

 force for the centre : in the next line is the ratio between the central force and that 



