An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics . y — i or x^ + (// — etn (nry~ = csc-«7r ; and (3) to ^?^+//^+2 e^^+l + 1 = 0 75 (5) (.r + ctnh ^vtt)-^ + y- = (6) (5) and (6) are circles. The circles (5) have their centres in the axis of Y,and pass through the points (— 1, 0) and (1, 0); and the circles (6) have theircentres in the axis of A. (4) is the complete solution, (6) is any equipotential line and (o) any line offlow for a plane sheet in which the points in the circumferen
An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics . y — i or x^ + (// — etn (nry~ = csc-«7r ; and (3) to ^?^+//^+2 e^^+l + 1 = 0 75 (5) (.r + ctnh ^vtt)-^ + y- = (6) (5) and (6) are circles. The circles (5) have their centres in the axis of Y,and pass through the points (— 1, 0) and (1, 0); and the circles (6) have theircentres in the axis of A. (4) is the complete solution, (6) is any equipotential line and (o) any line offlow for a plane sheet in which the points in the circumferences of two givencircles whose centres are further apart than the sum of their radii are kept atdifferent constant potentials, or where a source and a sink of equal intensityare placed at the points (—1,0) and (1,0). An important practical ex-ample is where two wires connected with the poles of a battery are placedwith their free ends in contact with a thin plane sheet of conducting figure shows the equipotential lines and lines of floAV of either system. The complete figure would have the axis of A for an axis of v=o If^„ v=i v,-^ v=o EXAMPLES. 1. Show that if /(.;•) = (?i when :f<i — h, f(.v) = a2 Avhen —h<^x b , r=^^+^ -\^(a,-a,) -ith tan-^ — 1- (c/2 — 3) tan-^ - 76 SOLUTION OF riiOBLEMS IX PHYSICS. [Akt. 47. 2. Show that if f(x)=() if ,v<0, f(x) = a, if () < .r i,/(.r) = ./., ifbi < X < h.^, f(x) = as if /, < .r ) tan~^ - + (>/., — a,) tan~^ ^ ./ U ./ + (3 — c-i) tan-^ — + 1/ 3. Show that if f(x) = - 1 if x < — 1. f(x) =x if — 1 < ./• l. r=i[(l+.)±^-(l-.)ta„^ + |log|l^^]. 4. Show that if f(x) = - 1 if .r < - 1, f(x) =0 if - 1< ,/? l, T- ir ,l±x ^ ,~|ttL // ?/ J Show that the equipotential lines are equilateral hyperbolas passing throughthe points (— 1, 0) and (1, 0), and that the lines of flow are Cassinian ovalshaving (— 1, 0) and (1, 0) as foci. The lines of fl
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