due to contamination of the potassium metaperiodate with periodate. The same effect was observed to occur whenever large concentrations of other organic solvents were used in solution. With lesser concentrations of alcohol or in water alone, the precipitated potassium metaperiodate seemed to be remarkably pure, considering that it was formed in the presence of large amounts of periodic acid and lithium ions.On the basis of these results, an analytical method for the determination of potassium was developed in which the potassium was separated as the metaperiodate from a 33% ethyl alcohol solution (4 ml. of water plus 2 ml. of ethyl alcohol). Although this same composition could have been established by trial and error means, the advantage of the method described above lies mainly in its convenience and the assurance that the result was positively determined. This method of graphical analysis can be applied also to existing solubility data, expressed as weight per volume of solvent mixture, provided the series of solubility data is sufficient to construct a solubility curve, plotting solubility against solubility composition. REFERENCES(1) Drude, F" Z. phyaik. Chem., 23, 267 (1897).(2) Jentoft, R. E., "Study of the Determination of Potassium as the Metaperiodate," thesis, University of Washington, 1952.

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AbstractsCanonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so-called Van Vleck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. I t will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. 'The Cayley-Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase-space. Explicit results have been obtained for several two-to .four-dimensional problems.Des transformations canoniques ont Ct C employtes frkquemment pour simplifier 1'Hamiltonien et d'autres optrateurs. Dans les thtories moltculaire et de l'ttat solide le dCveloppement de Van Vleck est gtneralement utilisC B cette fin, tandis que dans les thtories des particules intbragissant avec des champs on a prtfCrt une combinaison de transformations canoniques dans une forme close avec un type de dtveloppement de Van Vleck. Pour quelques-unes des transformations employkes dans les applications des formules en forme close sont bien connues. I1 est dtmontrC ici, que de telles formules peuvent Ctre obtenues quand la fonction de transformation est bilintaire dans les variables canoniques, et aussi qu'B l'aide des optrateurs matriciels il est possible de simplifier considCrablement les dkductions correspondantes. Avec le thCoreme de Cayley-Hamilton on peut exprimer les dtveloppements pour les optrateurs matriciels dans une forme close. Le nombre de termes d'optrateurs distincts ainsi obtenu est le mCme que le rang des matrices utilistes. Pour calculer les coefficients de ces termes d'opCrateurs on introduit un nouveau type de fonctions spkciales. Les transformations canoniques lintaires, qui en rCsultent comprennent des rotations gtntralistes dans l'espace ordinaire et dans l'espace des phases. Des resultats explicites ont t t t obtenus pour plusieurs problemes deux jusqu'8 quatre dimensions.Kanonische Transformationen sind oft verwendet worden um Hamilton-und andere Operatoren zu vereinfachen. In Molekul-und Festkorpertheorie wird gewohnlich die SOgenannte Van Vleck-Entwicklung zu diesem Zweck angewandt, wahrend in Theorien

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