proximate value of +እ -λ dx √(x - x')2 + a'} ' − a2} shall be a minimum for the whole length of the wire. In this way I find when λ is so 1 great that quantities of the order may be neglected, βλ A.231863-2 log aẞ+2aẞ; where .231863 &c. = 2 log 22 (4); (4) being the quantity represented by A in LACROIX' Traité du Cal. Diff. Tome 3, p. 521. Substituting the value of A just found in the equation We hence see that when the nature of the substance of which the wires are formed remains unchanged, the quantity aß is constant, and therefore ẞ varies in the inverse ratio of a. This agrees with what M. BIOT has found by experiment in the chapter before cited, as will be evident by recollecting that Blog μ'. From an experiment made with extreme care by Coulomb, 1 on a magnetized wire whose radius was inch, M. Bior has 12 found the value of ' to be .517948 (Traité de Phy. Tome 3, p. 78). Hence we have in this case which, according to a remark just made, ought to serve for all steel wires. Substituting this value in the equation (a) of the present article, we obtain g=.986636. With this value of g we may calculate the forces with which different lengths of a steel wire whose radius 1 is inch, tend to 12 turn towards the meridian, in order to compare the results with the table of COULOMB's observations, given by M. BIOT (Traité de Phy. Tome 3, p. 84). Now we have before proved that this force for any wire may be represented by It has also been shown that for any steel wire aß = .0548235, the French inch being the unit of space, and as in the present case a = 1 12, there results = .657882. It only remains there fore to determine K from one observation, the first for example, from which we obtain K= 58°.5 very nearly; the forces being measured by their equivalent torsions. With this value of K we have calculated the last column of the following table: The last three observations have been purposely omitted, because the approximate equation (a) does not hold good for very short wires. The very small difference existing between the observed and calculated results will appear the more remarkable, if we reflect that the value of B was determined from an experiment of quite a different kind to any of the present series, and that only one of these has been employed for the determination of the constant quantity K, which depends on f, the measure of the coercive force. The table page 87 of the volume just cited, contains another set of observed torsions, for different lengths of a much finer wire whose radius a= : hence we find the correspond 1 38 12 865 ing value of ß= 3,13880, and the first observation in the table gives K.6448. With these values the last column of the following table has been calculated as before: Here also the differences between the observed and calculated values are extremely small, and as the wire is a very fine one, our formula is applicable to much shorter pieces than in the former case. In general, when the length of the wire exceeds 10 or 15 times its diameter, we may employ it without hesitation. |