(c) Lines belonging to multiplets having f1 as initial orbits:-- The different combinations as shown by the arrows in the above diagram have been obtained by Catalán and Bechert. According to them the term P1 is uncertain. f, f, F1 are the biggest terms. Discussion of Results. According to the classification of the arc-spectrum of cobalt by Catalán and Bechert, f1, f, and F1 form the three closely successive highest groups of terms. Lines belonging to the fand f groups have come out prominently in absorption, while a few strong lines belonging to the F1 group have also been obtained. The absorption experiment definitely points to the fact that an f-term is the fundamental term of the cobalt arc-spectrum; but it is to be noted here that this is not in agreement with Bohr and Sioner's scheme of the periodic system, according to which a d-term is to be expected as the fundamental term. It is not possible to decide from this experiment whether for f is the fundamental term; for f corresponds to two close levels, the difference in these term-values being about 3500-1 cm., and, at the temperature necessary for the absorption experiment, transitions to the orbits corresponding to and f terms are equally probable, as is shown by the lines absorbed. A clue to this would probably be obtained by pushing the experiment further in the ultra-violet region. The lines belonging to the group f11 are intercombination lines, and Catalán and Bechert have suggested them to be the resonance lines; but only one line, λ=4190·709 (I.A.), of this group was obtained in absorption. This is the strongest line of the suggested system of resonance lines, and the other weaker lines lie on the longer wave-length side of this line. Photographs were taken in this region, but no other line could be obtained in absorption. It appears that the group of lines fo1 is not important enough to be regarded as the resonance lines. Summary. The paper contains an account of experimental work on the absorption spectra of aluminium and cobalt in the hightemperature furnace. The 2p1-2s, 2p-2s lines of Al do not come in absorption before a temperature of 1520° C. is reached. The higher members of ps- and pd-series are obtained in absorption at about 1700° C. The absorption spectrum of cobalt has been photographed from 4500 Å.U. to 3000 A.U. at a temperature of 2000° C. The lines have been identified by measurement, and tabulated according to the series classification of Catalán and Bechert. It is shown that many lines belonging to f1and ƒ2, and a few lines belonging to F1 terms, which are the three successive largest terms, have been obtained in absorption. Only one line, λ=4190-709, belonging to the group which Catalán and Bechert have suggested for the resonance lines has come out in absorption. We take this opportunity to express our sincere thanks to Prof. M. N. Saha for his guidance and interest in the work. XXXVII. The Determination of Young's Modulus from Compression Tests on Circular Cylinders. By F. É. RELTON, B.A., B.Sc., Building Research Station, Department of Scientific and Industrial Research *. THE present paper owes its genesis to the call for an explanation of certain anomalies in the behaviour of specimens under compressive loads. These anomalies, which were rendered visible by the use of translucent specimens, were more apparent than real, being nothing more than the fact that plane horizontal sections did not remain plane under load; but they served to show that the compression test, as usually carried out, did not necessarily give the true value of Young's Modulus, despite the refinements of physical measurement of which modern extensometers are capable. It has been claimed for these instruments that they are capable of measuring down to 10-5, and even 10-6, of an inch. It became a matter of interest, and of some practical importance, to inquire to what extent the method was in itself reliable as a means of determining the particular physical constant. If the claims put forth on behalf of the extensometers were valid, one could measure a movement to within 0.2 per cent. of its true value; but such refinement is futile unless the method can be made at least as reliable as this. Of the literature on the subject I have made most use of a paper by Prof. L. N. G. Filon †; of papers subsequent thereto I have consulted one by Dr. J. Dougall . The former of these contains a section devoted to historical references, and no further mention thereof will here be made. In those portions of Filon's paper which are relevant to the present discussion, the analysis is limited by the assumption that no movement takes place at the periphery of the end face of the test piece; afterwards a correction is made for the case where there is movement of known amount. In actual practice it would be exceedingly difficult to say precisely what the magnitude of such movement would be; I have therefore so analysed the problem that no such measurement is necessary. * Communicated by the Director. "On the Elastic Equilibrium of Circular Cylinders under Certain Practical Systems of Load," Phil. Trans. Roy. Soc. ser. A, vol. cxcviii. (1902). t "An Analytical Theory of the Equilibrium of an 1sotropic Elastic Rod of Circular Section," Trans. Roy. Soc. Edinb. vol. xlix. (1913–14). The present paper falls into three sections, of which two are quite short. The first section seeks to explain why plane sections do not necessarily remain plane after loading. The second section determines the conditions requisite for a correct evaluation of Young's Modulus and Poisson's Ratio. The third section seeks to give a method of determining Young's Modulus even when the conditions laid down in the second section are violated. Though the applications of the paper are Mechanical rather than Mathematical, the treatment can hardly be described as elementary, and as the third section is somewhat long an indication of its mode of development will be given here. It can be shown that when the displacements in a body are symmetrical about an axis, the strains and principal stresses can all be expressed in terms of a single function which satisfies a partial differential equation of the fourth order. The number of solutions to this equation is infinite, but two types are chosen (i.) solutions involving circular and Bessel functions, (ii.) a finite polynomial. From these a solution is formed by multiplying them by constants and combining them additively. This solution is then made to fit the requisite boundary conditions, thereby establishing certain relations among the constants. By a suitable choice it eventuates that the number of these relations is adequate to the solution of the problem. No cognisance is taken of the actual distribution of the load provided it be symmetrically disposed about the axis. The contraction of a known length and the extension of a diameter of the mid-section are computed; these can be measured by extensometers, and on these measurements the calculation of Young's Modulus is based. It appears in the sequel that the accurate determination of Young's Modulus is indissolubly bound up with a knowledge of the value of Poisson's Ratio. SECTION 1. In applying the Mathematical theory of Elasticity to the study of the behaviour of short cylinders in compression, there are certain obvious advantages in employing the cylindrical polar system of coordinates r, 0, z; where 2 is measured along the axis of the cylinder, r is measured from this axis. and from some axial qlane of reference. During a test, the load will, as far as possible, be applied symmetrically about the axis, and the resulting displacements may be regarded as symmetrical. This gives rise to certain simplifications of the fundamental equations: two of the three "rotations" of the polar element are obviously zero, viz., those about the axis of ≈ and about the horizontal (radial) axis, while differentiations with respect to disappear. The equilibrium equations are reduced from three to two in number. 2 If u, v, w denote displacements in the directions r, 0, z respectively we have v=0, and for the strains (in the usual notation) Certain experiments have been carried out on prisms of gelatinous material with coloured threads inserted. It was found that the threads, being originally horizontal, became curved as the prism was compressed. The explanation is as follows:-Working in cylindrical polar coordinates, without necessarily confining ourselves to symmetrical displacements, the assumption that horizontal plane sections remain plane and horizontal after loading is expressed analytically by the equations From equation (3), which gives the form of we, we have immediately ди дг = 0, which shows that the radial displace ment is independent of z and must be the same at all levels, including the end faces. The curving of the threads indicates that the free motion of the end faces was impeded. Phil. Mag. S. 7. Vol. 1. No 2. Feb. 1926. 2 A |