arrangement of the genera and classes is in accordance with the construction of Gauss, explained in the preceding articles; and the position of each class in the arrangement is indicated by placing opposite to it, in a separate column, the term to which it corresponds in the symbolic formula (such as K or ExK) which forms the type of the arrangement. To the two Tables of positive and negative determinants Mr. Cayley has added a third, containing the thirteen irregular negative determinants of the first thousand. In a letter addressed to Schumacher, and dated May 17, 1841, Gauss expresses a decided opinion of the uselessness of an extended tabulation of quadratic forms. "If, without having seen M. Clausen's Table, I have formed a right conjecture as to its object, I shall not be able to express an opinion in favour of its being printed. If it is a canon of the classification of binary forms for some thousand determinants, that is to say, if it is a Table of the reduced forms contained in every class, I should not attach any importance to its publication. You will see, on reference to the Disq. Arith. p. 521 (note), that in the year 1800 I had made this computation for more than four thousand determinants" [viz. for the first three and tenth thousands, for many hundreds here and there, and for many single determinants besides, chosen for special reasons]; "I have since extended it to many others; but I have never thought it was of any use to preserve these developments, and I have only kept the final result for each determinant. For example, for the determinant -11,921, I have not preserved the whole system, which would certainly fill several pages, but only the statement that there are 8 genera, each containing 21 classes. Thus, all that I have kept is the simple statement viii. 21, which in my own papers is expressed even more briefly. I think it quite superfluous to preserve the system itself, and much more so to print it, because (1) any one, after a little practice, can easily, without much expenditure of time, compute for himself a Table of any particular determinant, if he should happen to want it, especially when he has a means of verification in such a statement as viii. 21; (2) because the work has a certain charm of its own, so that it is a real pleasure to spend a quarter of an hour in doing it for one's self; and the more so, because (3) it is very seldom that there is any occasion to do it. . . . My own abbreviated Table of the number of genera and classes I have never published, principally because it does not proceed uninterruptedly."+ Probably the third of Gauss's three reasons will commend itself most to mathematicians who do not possess his extraordinary powers of computation. An abbreviated Table of the kind he describes, extending from -10,000 to +10,000, would occupy only a very limited space, and might be computed from Dirichlet's formulæ for the number of classes (see Art. 104), without constructing systems of representative forms. But it would, perhaps, be desirable (nor would it increase the bulk of the Table to any enormous extent) to give for each determinant not only the number of genera, and of classes in each genus, but also the elements necessary for the construction, by composition only, of a complete system of all the classes. For this purpose it would not be necessary to specify (by means of representative forms) more than 5 or 6 classes, in the case of any determinant within the limits mentioned. Mr. Cayley's Table of the first hundred negative determinants occupies about four pages of Crelle's Journal; the determinant -11,921 would occupy about one page. Briefwechsel zwischen C. F. Gauss und H. C. Schumacher, vol. iv. p. 30. Report on Observations of Luminous Meteors (ante, pp. 1-81). APPENDIX I.--Errata. (1) p. 35, December 8, Dundee. Column Appearance, &c. For A spearhead-like crescent moon, &c. read A spearhead; like crescent moon, &c. (2) p. 41, December 24, London. Column Direction, &c. Insert the words Radiant point Aldebaran. (3) p. 43, December 27, 8h 57TM P.M. ending, &c. read Track enduring, &c. Column Appearance, &c. For Track (4) p. 57, April 29, 11h 55m P.M. Column Appearance, &c. Read thus— Left no track. Brilliance vanished suddenly at b Lacertæ. Remaining 12° of the course light red (Mars at maximum robbed of his rays), very intermittent and vacillating, died out, 2-3 seconds. (5) p. 64, August 12, 11h 9m P.M. Column Position, &c. Omit the words short of the second. (6) From five accounts of the meteor 1862, September 19, the following is a calculation of its path : At London, after explosion overhead, the meteor proceeded a considerable distance towards 69° W. of N. At Nottingham the meteor passed sixty-three miles over London, seeking an earth-point 42° W. from S. At Hay (South Wales) the meteor passed fifty-seven miles over London, seeking an earth-point 70° E. from S. At Torquay the meteor passed 57 miles over London, seeking an earthpoint 9° E. from N. At Hawkhurst the meteor passed forty-seven miles over London, seeking an earth-point 66° W. from N. An earth-point seven miles S.W. from Hereford satisfies the observations in the following manner : London, 70° W. from N. (observed 69° W. from N.). (observed 42° W. from S.). The errors of observation being in no case greater than 5°, from the calculated bearings. A ground-point so close to Hay sufficiently explains anomalies in the observation at that place; but its distance is on the other hand 120 miles from London, where the meteor appears to have been fifty-six miles above the earth. The path of the meteor was therefore inclined downwards, from 25° above the horizon towards 70° W. of N. A visible flight of 115 miles, from eighty-three miles over Canterbury to thirty-three miles over Oxford, performed in three to four seconds of time, is the result obtained from the comparison of these observations. |