The present investigation deals with the invariant properties and criteria of special binary forms determined by given algebraic differential equations. It consists of two parts. The first part deals with the general methods that can be applied to the direct derivation of invariant criteria from the given differential equation. The results obtained help both in constructing a system of invariant relations whose validity determines whether a form with general coefficients can be transformed into the special form, and also they serve in actually setting up the transformation.

The second part is devoted to working out an especially interesting case of the general deductions of the first, namely for the differential equation of a spherical function, an investigation that I have undertaken at the urging of Professor Lindemann. As a result one obtains for a general spherical function of any degree and any order a sequence of invariant and simultaneous invariant relations using which one can answer the two fundamental questions, of the conditions for the possibility of transforming a form with general coefficients into a given spherical function, and the means of carrying out this transformation. The conclusion returns to more general points of view: on the one hand it describes the place of the special case of spherical functions within a more comprehensive class of problems, and, on the other hand, it contemplates a successful application of the principles of the first part to those more general questions.

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