An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics . the equi-potential lines above mentioned will be lines of , that is ^(J + tan-^) (5) reduces to U^~ ^ ^^^^ ^^ ? (^) If now we give to a values differing by a constant amount we get a set ofstraight lines radiating from the origin and at equal angular ^ = b, that is -iTT b, (6) reduces to x^-^,/^e-^\ (8) If we give to i a set of values differing by a constant amount we get a setof circles whose centres are at the origin and
An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics . the equi-potential lines above mentioned will be lines of , that is ^(J + tan-^) (5) reduces to U^~ ^ ^^^^ ^^ ? (^) If now we give to a values differing by a constant amount we get a set ofstraight lines radiating from the origin and at equal angular ^ = b, that is -iTT b, (6) reduces to x^-^,/^e-^\ (8) If we give to i a set of values differing by a constant amount we get a setof circles whose centres are at the origin and whose radii form a geometricalprogression. They are the equipotential lines for a thin plane sheet of infiniteextent where the potential function is kept equal to given different constantvalues on the circumferences of tAvo given concentric circles or where we have a source as the origin; and for thisi system the lines (7) are lines of flow, and (3) is the complete solution. The figure gives the equipotentiallines and lines of flow for either sys-tem, but only for positive values of complete figure has the axis of Xas an axis of Chap! IV.] FLOW OF ELECTRICITY IN AN INFINITE PLANE. 73 Solve the problem of Art. 44 for the case where /(.x)= —1 if x0. 2 :r A71S., V^ - tau^^ -. TT // 2. Solve the problem of Art. 44 for the case where f(x) = a if X 0. Ans., V= \ (<i -\-b) + -(b — a) tan^ - . 3. Reduce (7), (8), and (9) Art. 44 to the formsTry. ^+(A-a;) respectively. 46. An especially interesting case of Art. 44 is the following wheref(x) = 0 if x<-l, f(x) =1 if - 1< .r l. V=- ftan-1 ^^^^ + tan- ^^^^ . (1) Here Kow - log [(1 - z)ir\ = - log [(1 - X - ///)/] = - log [//+ (1 )*] l-log[()^ + /] + ^tan-^. and - ;^ log [(- 1 - ;^)/] = - _^ log [(-!•- yO C = - ^ log [y - (1+ x)q^-~ log [(1 + .ry + //] + ^tan- ^^.- log — = ~- log Vn , • + - tan- ^ h tan ^ . 74 SOLUTION OF PEOBLEMS IN PHYSICS. [Art. 4(j. Hence vrV // // / Jtt ° (1-\
Size: 2156px × 1159px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1890, bookpublisherbostonginncompany
