May 19, 1892. On Nova Auriga. By William Huggins, D.C.L., LL.D., F.R.S., and On the Changes produced by Magnetisation in the Length of Iron and F.R.S. On the Measurement of the Magnetic Properties of Iron. By Thomas On the Development of the Stigmata in Ascidians. By Walter Garstang, Page 486 495 503 505 Observations on the Post-Embryonic Development of Ciona intestinalis and Clavelina lepadiformis. By Arthur Willey, B.Sc. Lond. ......... 513 The Human Sacrum. By A. M. Paterson, M.D., Professor of Anatomy in University College, Dundee, St. Andrews University..... List of Presents............ 520 525 PROCEEDINGS OF THE ROYAL SOCIETY. March 3, 1892. Mr. JOHN EVANS, D.C.L., LL.D., Treasurer, in the Chair. A List of the Presents received was laid on the table, and thanks ordered for them. In pursuance of the Statutes, the names of the Candidates for election into the Society were announced, as follows: : Armstrong, Robert Young, Lieut.- | Gotch, Francis, M.R.C.S. Col. Beddard, Frank Evers, M.A. Beevor, Charles Edward, M.D. Blake, Rev. John Frederick, M.A. Boulenger, George Albert. Brennand, William. Buzzard, Thomas, M.D. Foster, Professor Clement Le Hill, Professor M. J. M., M.A. | Joly, John, M.A. Jones, Professor John Viriamu, Kidston, Robert, F.G.S. Larmor, Joseph, D.Sc. Love, Augustus Edward Hough, James Frederick MacMunn, Charles, M.D. Martin, John Biddulph, M.A. VOL. LI. B Matthey, Edward, F.C.S. Notter, James Lane, Surgeon- Smith, Rev. Frederick John, M.A. Stevenson, Thomas, M.D. Oliver, John Ryder, Major-General Tuke, Daniel Hack, M.D. Ulrich, Professor George Henry Veley, Victor Hubert, M.A. Worthington, Professor Arthur Young, Professor Sydney, D.Sc. The Right Hon. Spencer Compton Cavendish, Duke of Devonshire, a Member of Her Majesty's Most Honourable Privy Council, whose certificate had been suspended as required by the Statutes, was balloted for and elected a Fellow of the Society. The following Papers were read : I. "Certain Correlated Variations in Crangon vulgaris." By W. F. R. WELDON, M.A., F.R.S., Fellow of St. John's College, Cambridge, Professor of Zoology in University College, London. Received February 11, 1892. The first successful attempt to find a constant relation between the variations in size exhibited by one organ of an animal body and those occurring in other organs was made some three years ago by Mr. Galton; and in a paper read before the Royal Society (Roy. Soc. Proc., vol. 45, p. 135) he determined this relation between several organs of the human body. In what follows an attempt is made to apply Mr. Galton's method to the measurement of the correlation between four organs of the common shrimp. Before the details of the measurement are discussed, a short summary of the method will be given. Galton's starting point was the fact that each organ of a given race of men varies about its mean size to an extent and with a frequency If two A indicated by the probability equation (y = e-xe). π .c variable organs are known to vary in this way, and if they are so connected that when the deviation of one variable from its average is known the mean deviation of the second is known, then, evidently, a surface can, from these data, be constructed, showing the relative frequency of occurrence of all possible combinations between the two variables. The changes produced in such a surface by changes in the degree of interdependence of the two variables have been investigated, at Mr. Galton's request, by Mr. J. D. H. Dickson ('Roy. Soc. Proc., vol. 40, 1886, p. 63). The results of this investigation, which are of importance for the present purpose, are two :— (1.) In the population examined, let all those individuals be chosen in which a certain organ, A, differs from its average size by a fixed amount, Y; then, in these individuals, let the deviations of a second organ, B, from its average be measured. The various individuals will exhibit deviations of B equal to x1, x2, X3, ...., whose mean may be called am. The ratio am/Y will be constant for all values of Y. In the same way, suppose those individuals are chosen in which the organ B has a constant deviation, X; then, in these individuals, ym, the mean deviation of the organ A, will have the same ratio to X, whatever may be the value of X. (2.) The ratios am/Y and ym/X are connected by an interesting relation. Let Qa represent the probable error of distribution of the organ A about its average, and Q that of the organ B; then So that by taking a fixed deviation of either organ, expressed in terms of its probable error, and by expressing the mean associated deviation of the second organ in terms of its probable error, a ratio may be determined, whose value becomes +1 when a change in either organ involves an equal change in the other, and 0 when the two organs are quite independent. This constant, therefore, measures the "degree of correlation" between the two organs. A determination of this constant will now be made in the case of five pairs of organs of the shrimp. In accordance with Mr. Galton's notation, the constant will be denoted by r, the mean size of each organ by M, and the probable error of distribution about the mean by Q. The organs measured are shown in the woodcut fig. 1; they are: (1.) The total carapace length, measured in a straight line; (3.) The length of the sixth abdominal tergum; (4.) The length of the telson. The measurements made were recorded to within 0.05 mm., and arc expressed in terms of the body length, taken as 1,000. As the average length of the shrimps used was rather over 50 mm., the measurements, in the form in which they are recorded, are accurate only to the nearest unit. The roughness of the edges of the parts measured, and the fact that the animals were preserved in spirit, made any attempt to attain greater accuracy exceedingly difficult. The variations in the length of the four organs here discussed have already been shown to occur with a frequency which agrees very closely with that indicated by the law of probability (cf. Roy. Soc. Proc.,' vol. 47, p. 445). The closeness with which the distribution of deviations in the samples used agreed with that indicated by a probability curve may be gathered from the diagrams, figs. 2 and 3, which are fairly typical of the whole series. 1.-Total Length of Carapace and Length of Post-spinous Portion. The relation between these two parts has been determined in five races of shrimps; of these the sample containing the greatest number of individuals was obtained at the laboratory of the Marine Biological Association at Plymouth. The mean length of the carapace, in 1000 adult female shrimps from Plymouth, was 249-63 thousandths of the body length; the probable error of distribution about this average was 4:55. These numbers will be denoted in what follows by Me and Qe respectively. The mean length of the post-spinous portion (Mps) was 177.53, its probable error (Qps) being 3.50. These numbers having been determined, the individuals were sorted into groups, such that the recorded length of the carapace, c, was the same in each group. The mean length of the post-spinous portion, ps, was then determined in each group. The results are entered in the first two columns of Table I. Each pair of entries in the table gives a datum for determining the mean deviation of ps which is associated with a given value of c. If each value obtained in this way be divided by the probable error of the organ to which it belongs, a series of pairs of values will be obtained, from each of which |