1. Attention has been called by Rayleigh in a paper cited below to the ease with which the general nature of the aberrations of optical instruments can be discussed by means of a modified characteristic function. The object of the present paper is to work this method in more detail in order to get a complete account of the aberrations and to connect it with the known method of calculating the aberrations of a particular system. We write down the doubly modified function in 2, and in 3 deduce from it the singly modified function used in the rest of the paper. The meanings of the coefficients are given in 4. Attention having been called in 5 to some features that limit the usefulness of the method, a new expansion of more practical value is found in 6. The meanings of the coefficients of this expression are given in detail in 7, and some relations between them (
i. e.
between the spherical aberration, coma, etc.) in 8. The numerical calculation is discussed in 9. In 10 another method of calculating the characteristic function is sketched out.
The aberrations of a symmetrical optical instrument