A silicate of soda was formed by fusing together silica and carbonate of soda, in proportions indicated by the previous experiments— i. e. 23 silica to 54 carbonate of soda = 31 soda. The fused mass was crystallized. It was dissolved in water, and the solution evaporated in vacuo yielded a crystallized salt, which contained about 5 per cent. of carbonate of soda; when this, calculated as the ordinary 10-hydrated salt, was subtracted, the silicate agreed nearly with the formula NaO. SiO2+7HO. A crystallized salt of like composition was obtained when hydrated silica was dissolved in a solution of caustic soda, the silica and soda being in the same proportions, i. e. 31: 23. In these experiments it appears, therefore, that in the process of solution and crystallization a portion of soda is extruded. When exposed to a temperature of about 300° F., nearly all the water was driven off from these salts, less than 1 per cent. remaining. The fused silicate of lithia in like manner, when treated with water, appeared to split up into different compounds. After guarding himself from drawing any decided inference from the experiments recorded, the author concludes by observing that at present he can see no alternative but to admit of more than one equivalent for silicic acid (that is to say of more than one acid), the value of which is determined by circumstances, such as the presence of water and the nature of the base to which it is presented. The existence of such different silicic acids has been already suggested by chemists on different grounds, particularly by Ebelman and Laurent, and lately by M. Fremy. II. Appendix to the Paper of Mr. W. H. BARLOW, "On an Element of Strength in Beams subjected to Transverse Strain," &c. By PETER BARLOW, Esq., F.R.S. Received March 25, 1857. A notice of this communication is given in the abstract of the paper to which it refers (see antea, p. 432). III. "On the Application of Parabolic Trigonometry to the Investigation of the Properties of the Common Catenary." Some time ago, on the publication of a paper read by me last summer at Cheltenham before the Mathematical Section of the British Association on Parabolic Trigonometry and the Geometrical origin of Logarithms, Sir John Herschel called my attention to the analogy which exists between the equation of the common catenary referred to rectangular coordinates, and one of the principal formulæ of parabolic trigonometry. Since that time I have partially investigated the subject, and find, on a very cursory examination, that the most curious analogies exist between the properties of the parabola and those of the catenary,-that in general for every property of the former a corresponding one may be discovered for the latter. In this paper I cannot do more than give a mere outline of these investigations, but I hope at some future time, when less occupied with other avocations than at present, I may be permitted to resume the subject. I will only add, that the properties of this curve appear to be as inexhaustible as those of the circle or any other conic section. II. The equations of the common catenary referred to rectangular coordinates are The point O may be called the focus, whose distance to the vertex A of the curve is =m. It will simplify the investigations, without lessening their generality, if we assume the modulus or focal distance m=1. Assume 2 sec0=e+e-*, 2 tan 0=e*—e ̄* (1.) Now if we make x', x", "", &c. successively equal to 2x, 3x, 4x, &c., we shall have (see Parabolic Trigonometry,' No. XXVI.) Now (Parabolic Trigonometry,' No. III.) it has been shown that (sec 0+tan 0)"=sec (0+0+ to n terms) +tan (00- to n terms); hence in the catenary we shall have (y+8)=(y,, • . . n + 8,¡¡ . . . n), or if two points on the catenary be assumed, the abscissa of one being n times that of the other, the nth power of the sum of the ordinate and arc of the latter will be equal to the ordinate and arc of the former. We may graphically exhibit with great simplicity the sum of a series of angles added together by the parabolic or logarithmic plus. Let a set of equidistant ordinates—for simplicity let the common interval be unity-meet the catenary in the points b, c, d, k, l, and then let the catenary be supposed to be stretched along the horizontal tangent passing through the vertex A. Let the points b, c, d, k, l, on the catenary in its free position, coincide with the points ß, y, d, κ, λ, on the horizontal line when strained in that position, and as x or Aß, Ay, Ad, AK, AX is successively equal to m, 2m, 3m, 4m, &c. or to 1, 2, 3, 4, &c. if m=1, AOy such that, sec AOy+tan AOy=e2, or AOy=E+E AOd such that, sec AOd+tan AOd=e3, or AOd=ɛ±ɛɛ; consequently if we draw lines from the focus O successively to the points ß, y, d, к, λ, the angles AOß, AOy, AOƐ, AOK, AOX will be the angles, +ε, 8—8—8, ɛɛɛɛ. This is one of the simplest graphical representations we can have of angles added together by the parabolic or logarithmic plus. Hence as successive multiples of an arc of a circle give successive arithmetical multiples of the corresponding angle at the centre, so successive multiples of a given abscissa give successive arcs of the catenary which, extended along the vertical tangent, subtend at O successive parabolic multiples of the original angle. Since the original interval was assumed equal to m, and as the arc of the catenary is always longer than the abscissa which subtends it, or Aß>Ab, it follows, as has been shown in Parabolic Trigonometry,' No. XXII., that ɛ>45°. (e-e-*) we shall have =tan 0, but as is the dy dx dy dx trigonometrical tangent of the angle which the linear tangent at the point (xy) makes with the axis of the abscissa, hence this other theorem: Let a set of equidistant ordinates meet the catenary in the points b, c, d, k, l, and at these points let tangents to the curve be drawn, they will be inclined to the axis of the abscissa by the angles 0,00, 000, 0±0 ±0 ±0, &c., which is even a yet simpler geometrical representation than the preceding. Hence also it evidently follows that as the limit of the angle which a tangent to the catenary makes with the axis of the abscissa is a right angle, the limit of the angle 00000, ad infinitum, is a right angle. We have also this other theorem : If with the point ○ as focus, and A as vertex, we describe a parabola, and from the points ß, y, d, K, λ we draw tangents BB, C, JD, KK, AL to the parabola, the differences between these tangents and the corresponding parabolic arcs, namely, AD, AC, AD, AK, AL, will be m, 2m, 3m, 4m, or AB-6B=m, AC-yC=2m, AD-♪D=3m, AK-xK=4m &c. This is evident (see Parabolic Trigonometry,' No. XXVI.) for the angles AOẞ=ɛ, AOy=ɛ±ɛ, AOd=ɛ±ɛ±ɛ, AOk=ɛ±ɛ±ɛ±ɛ. We may further extend these properties of the catenary. To simplify the expressions, let Yo denote the ordinate of a point on the catenary at which the tangent makes the angle with the axis of X. Let So denote the arc measured from the lowest point, and let Xp signify the ordinate. Φ Now let x, x, x,, be the abscissæ of the three arcs whose tangents make the angles p, X, w with the axis of x, and let the equation of condition be simply x=x+x,. Then we shall have the following relations between the corresponding arcs and ordinates of the catenary— Let there be four arcs of the catenary whose abscissæ x, x, x 11 shall be connected by the following relation Let w, 4, x, be the corresponding angles made by the tangents to the extremities of the arcs Sw, Sp, Sx, S↓. Then we shall have the following relations between the arcs and the ordinates ST=S4YXY+SxY¢Y&+S¥Y¢Yx+SpSxS¥ |