Page:Encyclopædia Britannica, Ninth Edition, v. 11.djvu/622

588 HEAT / vdn = ff. Remark that (18) is obtainable from (15) by putting Q/47ra 2 = <r, and a = 00 ; or directly from (10) by integration over the plane. IX. "Linear Motion of Heat"; time-periodic plane-source; rate per unit of area, per unit of time, at time t, a sin Int : = - e~ nlx sin(2nt - n l -x - TT) where x> }- . (19). (2)i2w - +B **sin(2n<+ie-4>r) where o;<0 J (2w)*2 Verify that v satisfies - K*, which is .what (9) becomes <.7 dx* when v is independent of y and 2 ; also that = <r sin 2n when a; is infinitely small positive, and -f 2ir = ff sin 2nt when a; is infinitely small negative. X. "Linear Motion of Heat"; space-periodic simple harmonic solid source, with plane isothermal surfaces. Initial distribution, v = V sin ax, when t = Q. Solution for any value of t -P/ixt-i { - 4 - (20). P T" 00 r = V / dt siua(x - L /< "2^(^ Modifying the integral within the brackets to make it appear as an analytical expression belonging to the general theory of images, for the case of a single infinite row of images,- and equating the result to the right-hand member, we see that t 3 X i being any integer. It is obvious that for a we may substitute ja in the second mem ber, and in the factor * a(x - ) under the integral sign of the first member, without altering it elsewhere, j being any integer : thus we have 1 ^ -K ^t -) where S= I XI. "Linear Motion of Heat"; space -periodic arbitrary solid source, isothermals plane. Initial distribution, v=f(x), when t = 0, /denoting an arbitrary periodic function, period I ; so that/(ie + i7) =/(x), i being any integer. Two solutions (A), (B). (A) derived synthetically from (10) : where S= (24). (B) derived analytically and synthetically from (20). Find A , A v A 2 , ..., B I , B 2 , &c., by the harmonic analysis, to satisfy the condition ^ cos0 + A 2 cos20 + &c. "j 3 2 sin20+ &c. , where Theu-A + 2*6 * j (* > A; i" J HirX 2iir~r " - + B; sin J (25). (26). XII. Uniform row of simple instantaneous plane sources. Two solutions (A) and (B). (A), from XI. (A), (23) : (. (x+il) Axt (27). The No. 2 diffusion curve of 82 is the representation of the first term (i = 0) of this formula. (B), from XI. (B), (25) and (26) : v = * l + 2 2 ^ cos 2 - 77 ^ | (28). I ( j = l I The comparison of these two solutions is very interesting physi 1 This is Fourier s (i) of Art. 874 for the case of/(x) a periodic function. cally, and useful arithmetically. To facilitate the comparison, put <72(ir/c<)* = 7 and x/2(irKt^ =p (29) the two solutions become j=* i- Iv J? -*<P+W =1+2 ^2 e -rcos ^ . (30). The equation between the second and third member, virtually due originally to Fourier, is also an interesting formula of Jacobi s, Fundamenta Nova Theories Fundioniou Ellipiicarum, as was long ago pointed out by Cay ley. 2 Each formula is a series which con verges for every value of t however small or however great ; the first, (27), the more rapidly the less is t; the second, (2-8), the more rapidly the greater is t. For the case of t = l*/4itir, and x = Q (that is, q = 1 and p 0), the two series become identical. For the more com prehensive case of p = Q, but q unrestricted, the comparison gives the following very curious arithmetical theorem When <<^ 2 /4/c7r (or q^.1) the first solution (27) converges with so great suddenness that three terms suffice for most practical pur poses; when t^.P/4:Kir (or g^l) the second solution (28) converges with so great suddenness that one term (after the constant first term) suffices for most practical purposes. Thus by using the solu tion (27) for all values of t from zero to something less than l 2 /4Kw, and (28) for all values greater than the greatest for which (27) is used, we have an exceedingly rapid convergence and easy calcula tion to find v for any values of x and t. These formula?, thus used, have been of great practical value in calculating what is now known as the arrival curve of signals through a submarine cable, and in designing instruments to record it automatically and allow its telegraphic meaning to be read, or without recording it to allow its meaning to be read by watching the motions of a spot of light. It is clear that (27) and (28) express the potential at a point at distance x from one end of a cable of length ^/, at time t from an instant when a quantity ac of electricity has been suddenly communicated to that end of the cable, both ends being always kept insulated after that instant (as is done practically by an exceedingly short contact with one pole of a voltaic battery, the other being kept to earth). The value of v for x=l, and for all values of t from to <x> , represents the rise of the potential at the remote end towards the limiting value -, towards which the potential rises throughout the conductor. XIII. (X. in three dimensions.) Space triple periodic solid source ; in other words v= sin ax sin fiy sin yz, when t = 0. Solution for any value of t Remark that, as an analytical expression for the present case of the general theory of triply-multiple images, the triple integral within the brackets may be written 2jr 2ir 2ir a f P r y ,/,/,,,;r S ina(r- g) sin/3((/- >;) s y(z -?)-~, S - -, , where S= 22 2 e 4xt i, j, k, being any positive or negative integers. XIV. and XV. (X. and XI. in three dimensions.; The formuloe may be written down by inspection ; from L, with X. and XI. for guides. The analytical theorem thus obtained, corresponding to (30), in three dimensions, is interesting to pure mathematicians. XVI. Harmonic solutions. Any distribution of heat, whether in an infinite or in a bounded solid, which keeps its type unchanged in subsiding towards uniformity, when left without positive or negative sources, except such, essentially negative, as are required to fulfil a proper boundary condition, is called a harmonic distribu tion, provided the temperature does not increase to infinity in any direction. The boundary condition, if the solid is bounded, is essentially that the rate of emission from the surface at every point of it varies in simple proportion to the temperature, and at such a rate per 1 of temperature at each part of the surface as the solution requires. X. and XIII. are examples. The general con dition for a harmonic solution is /,=) .... .. .(34); 2 Quarterly Journal of Mattifmatics for ISjT ; Note by Caylcy on an article

by W. Thomson, entitled " On the Calculation of Transcendents of the form