cannot be represented by the simple formula C', and we must employ an expression of the form CXCXC To obtain an expression thus representing all the classes of the principal genus, we take for C, a class ap pertaining to the greatest exponent 0, to which any class can appertain; and in general for C, we take a class appertaining to the greatest exponent 8 to which any class can appertain when its period contains no class, except the principal class, capable of representation by the formula C,"1xCx.. C-- The number 0, x 0, x ... is called by Gauss the exponent of n third, &c., exponents of irregularity. From the mode in which the formula CxCx.. is obtained, it can be inferred that 0, is divisible by 0, 0 by 0,, and so on; whence it appears that a determinant cannot be irregular unless n be a divisible by a square; nor can it have r indices of irregularity unless n be divisible by a power of order r+1. Moreover, whenever the principal genus contains but one ambiguous class, the determinant is either regular or has an uneven exponent of irregularity; if, on the contrary, the principal genus contain more than two ambiguous classes, the determinant is certainly irregular, and the index of irregularity even; if it contain 2o ambiguous classes, the irregularity is at least of order, and the exponents of irregularity are all even. A few further observations are added by Gauss. Irregularity is of much less frequent occurrence for positive than for negative determinants; nor had Gauss found any instance of a positive determinant having an uneven index of irregularity (though it can hardly be doubted that such determinants exist). The negative determinants included in the formula, -D=216k+27, =1000k+75, =1000k+675, except -27 and -75, are irregular, and have an index of irregularity divisible by 3. In the first thousand there are five negative determinants (576, 580, 820, 884, 900) which have 2 for their exponent of irregularity, and eight (243, 307, 339, 459, 675, 755, 891, 974) which have 3 for that exponent; the numbers of determinants having these exponents of irregularity are 13 and 15 for the second thousand, 31 and 32 for the tenth. Up to 10,000 there are, possibly, no determinants having any other exponents of irregularity; but it would seem that beyond that limit the exponent of irregularity may have any value. 118. Arrangement of the other Genera.-In the preceding article we have attended to the classes of the principal genus only; to obtain a natural arrangement of all the properly primitive classes, we observe that, if the number of genera be 2", the terms of the product (1+T1) (1+r) (1+r)....... (1+r), in which I'; represents any genus not already included in the product of the i-1 factors preceding 1+Fi, will represent all the genera. If, then, A1, A2, A represent any classes of the genera I',, ',,.. F, respectively, and C be the formula representing all the classes of the principal genus, the expression |K=|C|× (1+A,) (1+A)... (1+A,) supplies a type for a simple arrangement of all the classes of the given determinant. When every genus contains an ambiguous class, it is natural to take for A,, A,,.. A, the ambiguous classes contained in the genera I1, I respectively. When the principal genus contains two ambiguous classes (and when, consequently, one-half of the genera contain no such classes), let C, be the class taken as base (or, if the determinant be irregular, as first of the bases) in the arrange ment of the classes of the principal genus, and let 2-C,; it may then be shown that, will belong to a genus containing no ambiguous class, and that the formula K |C|× (1+0) (1+A)... (1+A), in which A,,.. Au, are ambiguous classes, represents all the classes *. In general, if the principal genus contain 2 ambiguous classes (a supposition which implies that the determinant is irregular, having к even exponents of irregularity, and that there are only 2- genera containing ambiguous classes)-let '-C,; Q2=C, ; . . .Q, 2=C-it will be found that all the classes are represented by the formula K|C|× (1+,) (1+0).. (1+Qx) (1+Ax+1).. (1+A), in which Ax+1,... A, are ambiguous classes, and 2,, ... classes belonging O 1 to genera containing no ambiguous class †. = A similar arrangement of the improperly primitive classes (when such classes exist) is easily obtained. Let 2 denote the principal class of im D-1 properly primitive forms, i. e. the class containing the form (2, 1, −D1); 2 we have seen (Art. 113) that the number of properly primitive classes which, compounded with 2, produce 2, is either one or three. When there is only one such class, the number of improperly primitive classes is equal to that of properly primitive classes; and if K be the general formula representing the properly primitive classes, the improperly primitive classes will be represented by K. When there are three properly primitive classes, which, compounded with 2, produce 2, the principal class will be one of them, and if o be another of them, 2 will be the third; also and 2 will belong to the principal genus, and will appertain to the exponent 3. When the determinant is regular, instead of the complete period of classes of the principal genus, 1, C, C2, . . Cn-1, we take the same series as far as the class C exclusively; when the determinant is irregular, we can always choose the bases C1, C,, . . in such a manner that the period of one of them shall contain Φ and 2, and this period we similarly reduce to its third part by stopping just before we come to or p2. Employing these truncated periods, instead of the complete ones, in the general expression for the properly primitive classes, we obtain an expression, which we shall call |K', representing a third part of the properly primitive classes, and such that 2xK' represents all the improperly primitive classes. 119. Tabulation of Quadratic Forms.-In Crelle's Journal, vol. lx. p. 357, Mr. Cayley has tabulated the classes of properly and improperly primitive forms for every positive and negative determinant (except positive squares) up to 100. The classes are represented by the simplest forms contained in them; the generic character of each class, and, for positive determinants, the period of reduced forms (Art. 93) contained in it, are also given. The Gauss employs a class 2, producing C, by its duplication, both when one and when two ambiguous classes are contained in the principal genus. The number of classes requisite for the construction of the complete system of classes is therefore μ in either case, since C, may be replaced by 221. + The principles employed by Gauss for the arrangement of the classes of a regular determinant are extended in the text to irregular determinants. If the determinant have ' uneven exponents of irregularity, the number of classes requisite for the construction of the complete system of classes is μ+K'. The simplest form contained in a class is that form which has the least first coefficient of all forms contained in the class, and the least second coefficient of all forms contained in the class and having the least first coefficient. If a choice presents itself between two numbers differing only in sign, the positive number is preferred. In the case of an ambiguous class of a positive determinant, the simplest ambiguous form contained in the class is taken as its representative. 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 2xK) 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 formula 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 573 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. |