U1= If now we write, in accordance with (37), Vi (45) hk)=-rl+farlo(h— k) +fabrlohk + zaslhk-1}oa?r15 (1 — k)2 + goa?rlhk. . (46) B=r-arl?(—k)- jabrlohk- jasl’hk- jarlhk +34a?r14 (12 — k) – a?rlohk=–2sl—r— jarl + jabrl(h—k) + fast* (h — k) + šarlo(1. – k) + farl* (h—k). (47) C=8+ {alę,— fabrl? (h— k) — 2asl? (le -- k) - arl(h-k) - Laorl+ (1 - k) =-3tl – 2s-abrl - Pasl - Barl— 18arl. . (48) D=t+abrl + past + farl + a?r24. (49) Moreover, since (38) may be written in the form u1(1+n'?) + 2gn=(A + Bn + Cm? + Dnö)o(1+ahkn +a(h— k)m2 — an) +2gn=constant, (50) we readily obtain 2AB+ahkA? +2g=0, (51) 2AC+B? +ash-k)A?=0. (52) 2AD+2BC-aAa=0. (53) From the equations (46), (47), (48), (51), (52), (53), the six quantities 9, r, s, t, a, and b may be calculated, and if we had retained everywhere terms of one higher order, we might have got eight equations with eight unknown quantities, &c. By a first approximation we readily obtain from (46)–(49): 9 q 12 1 - aq q1 A=q; B= =-1; C= * -saql; D=-* +žaq-zabql + £a?yl® ; and then from (51)-(53), 3 q=gl ; a= js ; b= 41 (54) . . 3 . Proceeding to the second approximation, we find 9 hk 12 I 19 B= ? h-k 9 ; 8 12.7 and then again from (51) and (52), h-ki 3 15 h-k (55) ge=gb(1+"7“); ; a= By means of these results we may now readily obtain from (1) and (2) expressions for u and v including respectively the terms of the 2nd and 21th order. They are :h-k 3 (h — k)? 33 hk h+ n + (1+ 21 20 12 40 12 U= where VI. Calculation of the Equation of the Surface. We will now show how for the equation of the surface of a stationary train of waves a more correct expression than (20) can be deduced. For this purpose we have to integrate the differential equation (39), or rather we have to prove that a series can be given which solves this equation to any desired degree of accuracy. Now such a series may be obtained in the following manner. Let hi (60) . represent the solution of an equation (61) where h; and ky have values which are slightly different from those of h and k in (39); then these values and the coefficients A, B, &c., of a series η= αη, + βη,° +γη,3 + δη +... (62) may be determined in such a way* that this series (62) satisfies the equation (39). Indeed, substituting (62) in (39) and taking into account (61), equation (39) reduces to (a +237 +3ryn,? + ...)?(hın)(ki+n) = (a +Bnityn.? + ...)(h – anı-Bmi? - Yni + ...) (k + ami +Bnia +ynil + ...)(1+banı +(bB+ca)ni? + ...), and it is only necessary to equalize the coefficients of the corresponding terms of both members of this equation. If we retain all terms to the fourth order, we find in this way, after some reductions :ahıkı-hk=0). (63) a? (hı - kı)-a?(h— k) – (ba?-38)hk=0.. (64) -a3 +24—(ba* — 222B)(h-k)-(ca’ — 2ba2B+882— 5ary)hk=0 (65) -4aß +3a2B + ba? — (ca? + 3ba?ß—382 — 4ary) (h-k)=0 (66) -483–6ay+ ca'+4bao8+3a8o+32°y=0. :. (67) To a first approximation these equations are satisfied by taking h=h; kı=k; a=1; B=b; y=x2 + }c (68) If then we substitute in (63), (64), (65), and (66) ho=h+€, kq=k+a, a=1+Q1, B=b+Bi where a, and B, are quantities of the first, e and æ of the second order, we find from these equations by second approximation: . . . 1 * The coefficient a in (61) might also have been chosen slightly different in value from a in (39), but this would only have introduced an unnecessary indeterminateness in the solution, e=-bhk; a=bhk; a,= -1(1—k) ; Bi=(-202 + 3c) (h-k). . (69) Substituting as a third approximation : h=h—bhk +€1; ki=k+bhk +@n, a=1-1(h-k) + az, we obtain finally, €1=}chk(-h+2k); R = {chk(2h —k); 42=(12— 3c)(k? — hk + ko).(70) Hence the equation of the surface of the waves is, including all terms of the third order : n=[1–6(1 — k) +(62—c)(ha-hk + k?)] + [6+(-212 + }c) (h-k)]n? +(12 + 3c)n +. (71) where hu htki 3 (73) 47 If we confine ourselves to that degree of approximation for which all the calculations have been effected, we may write for the equation of the wave-surface : 3(k 3 21+ (74) 3(h+k) (75) h (76) h+k For the solitary wave, when k=0, we have * 3 n n = (77) 41. + η= . * Another close approximation of the surface-equation of this wave has been deduced by McCowan, Phil. Mag. [5] vol. xxxii. (1891), p. 48. XLII. The Tin-Chromic Chloride Cell. By S. SKINNER, M.A.* А CELL consisting of a tin and platinum couple in a solution of the green chromic chloride was described by W. E. Case in the Proc. R. S. 1886, p. 345. The tin is dissolved by the chromic chloride at a high temperature only, and when the solution is cooled the tin is precipitated. The chemical changes are represented by the following statement : Cr,Cle+Sn SnCl,+2CrClz. : As the reaction is reversed on cooling, the cell has the interesting feature that at the end of a hot and cold cycle it is in the same chemical condition as it was at the commencement. It thus offers a method of deriving electrical energy directly from heat. The author of the account gives a curve of electromotive force, and finds that it is zero at 60° F., and increases to about 1 volt at 200° F. I shall show that the E.M.F. is not zero at the ordinary temperature, but is about volt ; however, the cell will not give any current at these temperatures on account of polarization. It appears that the curve given by Mr. Case is not an E.M.F. curve, but one which was probably obtained by using a wire voltmeter, and therefore really represents the current the cell is capable of producing The tin is precipitated in small crystals from the cooling solution of Crēl, and SnCl2, and does not then form a satisfactory electrode. I have therefore arranged the cell with an amalgam of tin in place of the solid rođ. The precipitated crystals fall into the mercury and dissolve so as to reconstitute a suitable electrode. My construction of the cell is very much like that of a Clark cell. A test-tube with a platinum wire through the base has fluid amalgam in it, and this is covered with a solution of pure Cr,Clo, made by dissolving violet sublimed Cr,Clo in water with the aid of a fragment of tinfoil. A platinum plate and wire form the positive pole of the cell. Connecting such a cell at 15° C. with a galvanometer, there is a sudden deflexion which very rapidly becomes less until some small steady value is reached. On warming the cell the deflexion increases until it is relatively large. These observations show that the cell cannot produce a continuous * Communicated by the Physical Society: read February 8, 1895. |