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the effort of the free state people to protect themselves against a fradulent government, elected by the citizens of a foreign state.
At the close of Major Abbott's address Mr. B.W. Woodward made a few appropriate remarks comparing the Kansas contest to the struggle against English oppression in 1774.
F. H. HODDER.
A Kansas Woman's Work in Mathematics.
The last number of the Annals of Mathematics contains an article filling sixty-three quarto pages entitled: "Concomitant binary forms in terms of the roots," by Miss Anna L. MacKinnon. This dissertation was presented to the faculty of Cornell University in partial fulfilment of the requirements for the degree of Doctor of Philosophy.
Miss MacKinnon graduated from the Kansas University in the class of '89. After a year of graduate work and two years spent in teaching mathematics in the Lawrence High School she entered Cornell University. At the end of her first year she won the Fellowship in Mathematics and at the end of her second year received the degree of Doctor of Philosophy. She was then awarded the traveling fellowship of the Collegiate Alumnae Association, and for the last two years has been studying at the University of Goettingen under Professor Klein.
This thesis is a valuable addition to our knowledge of the Modern Higher Algebra, or the Invariant Theory as it is now called. After fifty-five years of cultivation by an army of giants this field of research is not yet exhausted. All the results of their labor, except the most recent developments, have been collected and presented from different points of view in five treatises, two in English, two in German and one in French. The invariants and covariants of binary forms may be expressed either in terms of the coefficients or in terms of the roots of the forms. Much stress has been laid upon the first mode of expressing the results, and complicated analytic machinery has been devised and perfected for performing the
calculations. On the other hand the expression of the results in terms of the roots has been correspondingly neglected. Here and there are to be found invariants and covariants expressed in terms of the roots; but there has not no general recognition of the possibility with a mode of expression and no attempt to general theory. Herein lies the value of MacKinnon's work. She has developed general theory of the invariant forms in terms of roots and devised the analytic machinery necessary for the calculations. Her new symbism springs naturally out of the symbolism of Clebsch and Gordan, which so many English speaking mathematicians have so studiously neglected.
One or two minor defects ought to be mentioned. Although Klein's now famous Erlangen Program is given in the list of references, there seems to be no acquaintance with its spirit. There is no hint anywhere in the paper of the relation of the old Invariant Theory to the modern theory of groups. Given a manifoldness and a group of operations on that manifoldness, to develop the invariant theory of the group; such is the modern statement of the problem. The invariant theory of the general projective group in one dimension (the theory treated in the paper) is only one out of many invariant theories, as Lie has so well shown. After a two year's residence in Goettingen Miss MacKinnon would now probably treat Semi-invariants and Semi-covariants as the invariants and covariants of the sub-groups of the general projective group. The blame for these deficiencies should be charged to her American instructors rather than to Miss MacKinnon herself.
The paper as a whole is exceedingly creditable to American scholarship. It gives evidence of extensive reading and shows that the writer has mastered her subject from all points of view except the one pointed out above. No American woman has done a better piece of work in the science of mathematics than this dissertation by Miss MacKinnon. H.B.N.