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which appear as straight lines, alternately dark and light, perpendicular to the line joining the mirrors.
As the outer mirrors are separated, the visibility of the zero fringes remains constant, while that of the interferometer fringes gradually decreases until a point is reached where the latter vanish; as the mirrors are further separated the interference fringes should reappear, approach a secondary maximum, and again disappear, this phenomenon being repeated indefinitely.
We are interested chiefly in the first disappearance. The distance between the mirrors corresponding to extinction is the observed quantity required to determine the angular diameter of the star. Thus far it has been necessary to move the mirrors by steps, as screws are only now being installed to maintain the mirrors at equal distances.
The visibility of fringes obtained with the interferometer pointed on Vega in August, 1920, with the mirrors separated 18 feet (5.5 m.), was fully as great as with the mirrors 6 feet (1.83) apart, thus indicating that atmospheric conditions will easily permit the use of even greater separations.
Calculations of stellar diameters based on estimates of surface brightness made by Eddington, Russell, and Shapley, indicated that a Orionis, because of its relatively large diameter, would be a promising object to attempt to measure with the 20-foot interferometer. Merrill first examined the star with the apparatus used by Anderson2 in the measurement of Capella and found a definite decrease in visibility for the maximum separation of the slits (100 inches). This was true for all position angles, thus indicating that the star was not a binary.
On December 13, 1920, the interferometer was tested by Pease on B Persei and then on y Orionis. Both stars are known to have diameters much smaller than can be measured with this instrument. When adjusted with the mirrors separated 121 inches (229 cm.), both the zero and the interferometer fringes appeared in the eyepiece. When the instrument was directed to a Orionis the interferometer fringes could not be found. a Canis Minoris was then observed and the interferometer fringes were easily seen thus indicating that the instrument was still in adjustment. It is thus clear that the disappearance of the fringes for a Orionis was real and not due to any disturbance of the mirrors, for the seeing was good and any flexure changes involve only a very slight shift of the compensating wedge to bring the fringes into view again.
On December 14 and the nights following, the seeing was poor and as the visibility of the zero fringes was decidedly lower than on the night of the 13th, no attempt was made to work on a Orionis. With a mirror separation of 13 feet attention was directed to a Ceti, a Tauri and ẞ Geminorum. The zero fringes were seen in every case, though much reduced in visibility, but as the interferometer fringes could not be seen at all except at intervals of better seeing, it is presumed that there was an actual de
crease in visibility and that with further observation some estimate can be made of the diameter of these stars.
Assuming that the effective wave-length of a Orionis is 5.75 X 105 cm. and that the value of d is 121 inches (306.5 cm.), the angular diameter of a Orionis from the formula a = 1.22 proves to be 0.047. An estimate of its linear diameter may be made by using a mean parallax of 0".018, which gives a diameter of 240 X 10 miles, or slightly less than that of the diameter of the orbit of Mars.
Corrections to this value will be derived by an experimental determination of the value of λ for this particular star; by a more accurate setting of the mirrors, for the uncertainty of this measure is at least 10 per cent; and by further determination of the parallax. The angular value given above is that corresponding to a uniformly illuminated disk. A darkening toward the limb, equal to that of the sun, would require an increase in the diameter of about 17 per cent.
We wish to express our obligations to Director Hale both for his encouragement and for placing the resources of the observatory at our disposal, and to Mr. J. A. Anderson for his checking of the measures on the night of December 13.
These PROCEEdings, 6, 1920 (474-475).
Mount Wilson Contributions, No. 185; Astroph. J. Chicago, 51, 1920 (263-275). "The weighted mean of Adam's spectroscopic parallax, 0.012 and the trigonometric parallaxes of Elkin, 0.030, and Schlesinger, 0.016.
AN OVERLOOKED INFINITE SYSTEM OF GROUPS OF
By G. A. MILLER
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS
Communicated April 28, 1921.
The determination of all the possible abstract groups of order pq2 p and q being distinct prime numbers, was considered by Cole and Glover in an article published in the American Journal of Mathematics, vol. 15 (1893), p. 191 and by O. Hölder in a long article published in the Mathematische Annalen, vol. 43 (1893), p. 301. In a subsequent article, published in volume 46 of the latter journal, Hölder directed attention in foot-note on page 323 to the fact that the enumeration of these groups contained in the former of the two articles mentioned above was incomplete.
The main object of the present article is to establish the fact that there is an infinite system of abstract groups of order pq2 which was overlooked not only by the authors already mentioned but also by others, including W. Burnside who gave an incomplete list of these groups in both editions of his well known and meritorious work entitled, "Theory of Groups of
Finite Order" 1897 and 1911. The system of groups in question consists of q-1 distinct groups for all values of p and q which satisfy certain conditions to be noted later while only one such group is given in the published lists. This system of groups is also interesting in view of the fact that each group of the system contains q characteristic operators including the identity.
To construct the groups of the system in question suppose that q is a divisor of p-1 and establish a simple isomorphism between q cyclic groups of order pq written as regular substitution groups. Let t represent the substitution of order q which permutes the corresponding letters of these q cyclic groups so that t is commutative with each of its substitutions and together with the cyclic group of order pq formed by the given isomorphism generates a regular abelian group of order pq2. Let S, S2..... S1 be substitutions of order q and of degree pq-q which transform corresponding generators of the given cyclic groups of order pq into the same power belonging to exponent q modulo pq and so chosen that the product S, S...S is commutative with t. Finally let So represent a substitution of order q contained in the first one of the q given cyclic groups of order pq. The product SS1S2...Sqt is a substitution of order q2 whose qt power is the substitution of order q, in the group formed by means of the said isomorphism, whose constituent is So. Hence it results that the pq substitutions of the group of order pq2 thus constructed which transform into a given power belonging to exponent q modulo pq a generator of the given cyclic group of order pq can be so chosen that their qth power is an arbitrary operator of order q contained in this cyclic group. The totality of these pq substitutions must correspond to itself in every automorphism of this group of order pq2. Hence this qth power must be a characteristic operator of the group.
From the preceding paragraph it results that each of the groups of order pq2 under consideration contains q characteristic operators including the identity and that any of these operators which is of order q can be made the qth power of all the pq operators of the group which transform the operators of order p in the group into a particular power. Hence there are q-1 distinct groups for particular values of p and q which satisfy the conditions that p and q are such primes that p-1 is divisible by q. These q-1 groups are conformal: that is, they contain the same number operators of each order. It is well known that for any prime number of q there is an infinite number of prime numbers p such that p-1 is divisible by q and hence there is no upper limit to the number of such distinct conformal groups.
The smallest order for which there exist at least two such conformal groups is 63. In this special case one of the characteristic operators of order 3 is the third power of the operators of order 9 which transform the operators of order 7 into their fourth powers while the other operator of order 3 is the third power of those which transform the operators of
order 7 into their squares in one of the two conformal groups. In the other group the reverse is true. Hence it is not possible to establish a simple isomorphism between the operators of these two groups. In view of the elementary properties of these groups it appears strange that the wide-spread error noted above was not corrected for more than a quarter of a century, especially since the incorrect results obtained by O. Hölder have been used by various writers in extending his work.
THE GLOBE, A SIMPLE TRISOMIC MUTANT IN DATURA
STATION FOR Experimental EVOLUTION, COLD SPRING HARBOR, N. Y.
In a series of articles already published, 1, 2, 3 or at the present writing in press (American Naturalist, Genetics), a number of recurrent mutants discovered in the Jimson Weed (Datura Stramonium) have been described and their peculiarities in external appearance shown to be connected with the presence of one or more extra chromosomes in their nuclei. Evidence has been presented which indicates that a given mutant of the "simple trisomic" type is conditioned by the presence of a single extra chromosome in a specific one of the 12 chromosomal sets. Such a form is called a simple trisomic mutant since in its somatic nucleus one of the 12 sets is a trisome with three homologous chromosomes instead of all the sets being disomes with two chromosomes each. The presence of an extra chromosome in a specific chromosomal set not only causes specific peculiarities in the growth and appearance of the mutant which results, but also brings about peculiarities in the inheritance of the mutant complex.
It is the purpose in the present paper to summarize the findings in regard to one of the simple trisomic mutants-the Globe-in anticipation of a more detailed paper to be published shortly in Genetics. The data were accumulated for the most part before the chromosomal condition in the Globe and other simple trisomic mutants had been determined by my colleague, Mr. John Belling, from studies of mitotic figures in the pollen mother cells. The Globe was the first mutant recognized in the Jimson Weed, having been discovered in 1915. Its depressed globose capsules suggested the name. Its adult characters as well as the broad entire leaves of its seedlings render the Globe one of the easiest mutants to recognize at any stage of development. It is the only one in fact that we have been able to pick out readily in the seed pan. Since usually it has not been necessary to grow plants beyond an early seedling stage when it is desired to distinguish Globes from normals, it has been possible with this mutant to base conclusions on a larger number of individuals than could readily have been obtained if we had been dealing with the other mutant forms.
The Globe mutant differs from normals apparently in all parts of the plant. It shows a complex of characters readily recognized, whether the plants in question have purple or white flowers, many or few nodes, and spiny or smooth capsules. Globes, like other mutants of this type, are slower in growth than normals; and in competition with normals are liable to be crowded out by them.
Globes Selfed and Crossed Reciprocally with Main Line NormALS. PLANTS POLLINATED IN GREENHOUSE, 1916-17. RECORDS TAKEN IN SEED PANS IN GREEN
Percentage difference between offspring from Globe Selfed and from Globe x normal =3.39 0.935. Diff./E. Diff. = 3.59.
Early breeding work with the Globe indicated that the manner of its inheritance was not in accord with any simple Mendelian interpretation. A more extensive series of selfs and crosses was therefore carried out in order to determine more accurately the extent to which the Globe complex could be transmitted. In table 1 is presented a summary of the experiment. In this table are given the number of seeds planted and the percentage of recordable seedlings which they produced. It is obvious that a difference exists between the inheritance through the male and that through the female parent. The egg cells of Globes, whether fertilized by Globe pollen or by pollen from normal plants, transmit the Globe complex to only about 25 per cent of the offspring. Globe pollen, when used on normal plants, transmits the character to only about 3 per cent of the offspring This figure, though small, is too large to be accounted for by the occurrence of new Globe mutations, in view of the rarity of Globes from normal parents in comparable material. The normal offspring of Globes seem to be true normals so far as their breeding behavior is concerned.
If 3% represents the average number of Globe pollen grains which transmit the mutant character, we should expect a higher proportion of Globe offspring when Globe pollen is used than when pollen from normals is used on Globe females. The reverse, however, is actually the case, and Globes selfed produced a lower percentage of mutant offspring than Globes pollinated by normals. The explanation may possibly be connected with a difference in vitality between seedlings produced by self and those produced by cross pollination. The fact is at least suggestive that a lower per