206 Prof. G. M. Minchin on the Magnetic Field Let PN be the perpendicular on the plate from P; let AB be the diameter in which it is cut by the plane through the axis of the plate and P; let Q, S be two very close points on the circumference, NQ making the angle 0 with NA. Then we shall suppose the plate broken up into triangular strips such as QNS, and calculate the potential of each strip at P. Let L be any point on NQ, let NL-; then the poten mžddd édé tial of the element mędę do at P is and the potential, V, of the whole plate is 2m√(√2+2)də.. П 2 Now if, as in the figure, the point N falls within the plate, the limits of are 0 and 7; if N falls on the edge of the plate, at B, the limits are 0 and and if it falls outside the plate, the limits are 0 and 0. Taking as the independent variable would, then, give us three different expressions for V, according to the position of N; and hence we must choose a more convenient variable than 0. Let & be the angle QOA, and change the expression (2) into one in which is the independent variable. We shall then have V m 0 0 D do and if the distances PA, PB are denoted by p, p', respectively, we have -=-2πz+2рE+(a2x2)K+2 − 2πz + 2μE + 2 (a2 —¿2)K+2 P where E and K are the complete elliptic integrals of the The integral in (5) is the complete elliptic integral of the 4ax This parameter is numerically greater than the modulus; and we converted into one with parameter n, the parameter in (5), k2 n If the angles PBA and e', respectively, we see that k2 so that the new II (n, e) + II(**, e)=√ tan-1 (√atan €)+K(e), where e is the amplitude of each of the two functions of the third kind (denoted by II), and a=(1+n)(1+ for complete functions (e=) we have (() = π 2√a sin2 = n Hence πpa+x+K-II(—cos2 01), ・ ・ (6) 2za-x and (5) becomes V == m pa+a This expression holds without ambiguity for all positions of the point P, and it shows that for all points so that the points, occupying any of these positions, at which V has any assigned value can be easily found. Thus, to find the point on OV at which this point Hence draw below AB, parallel to it and at the distance 2π' a right line, meeting VO produced in O'; then the perpendicular to AO' at its middle point meets OV in the required point. Now every complete elliptic integral of the third kind can be expressed in terms of complete and incomplete functions of the first and second kinds. Thus, for a complete function with the parameter -m it is known that, if we put m= — k'2 sin2 e, where A stands for √1-k'2 sin2 e, and K', E stand for incomplete functions of the first and second kinds with modulus k' and amplitude e. In the present case, V 2m = = 2{KE, + EK', — KK-T}+pE+p' cos e cos '. K. (14) It is evident that we may define the position of any point, P, in the plane of the figure by means of the two coordinates k and 0. Thus we have {E+k' sin 0 (KE', +EK'—KK'。—π) + cos 0. KA}.. (15) This, then, is the expression for the potential at any point in terms of the coordinates (k, e) of the point. In particular, it gives the value (9) for any point on the perpendicular through B to the plate, since for such a point 0=, and then the coefficient of 'sin within the brackets 2' π is equal to -, by Legendre's well-known relation between 2' the complete complementary functions, viz., π KE'+EK'-KK'= 2' whatever the modulus k may be. From (14) we can derive an expression for the conical angle subtended at any point, P, in space by a circle, i. e., for the magnetic potential due to a current coinciding with the circle. It is well known that this conical angle is numerically equal to the component of the attraction, perpendicular to the plate, at P due to a uniform circular plate coinciding with the aperture of the circle-a result which is evident from the principle that the current can be replaced by a magnetic shell, or thin plate, the upper and lower surfaces of which are, of course, of opposite signs. But the resultant potential of these two indefinitely close plates is the difference between the value of V in (14) and the value which (14) assumes when z+Az is substituted for z; that is, the magnetic potential at P due to the current is Az, and the strength of the dV dz magnetic shell is m . Az, which is i, the current in the circle; so that the magnetic potential is i multiplied by minus the differential coefficient of the right-hand side of (14) with respect to 2. Denote the function + KK-KE,- EK, by the symbol A, and for simplicity in the differentiation with respect to (x being constant) write (14) in the form and, regarding P as determined by the coordinates (z, x) instead of (k,), we have |