The paper deals with the mathematical theory of integral dose in radium therapy in which the study of point sources is of prime importance.It is shown that if all the γ-ray energy is absorbed 1 mg. hr. is equivalent to 3800 gm. r. It follows that when all the energy is absorbed 1 m.c.d. is equivalent to approximately 0·5 megagm. r or 1 calorie. The theory of integral dose in a sphere due to a point source is then developed and formulæ given for its value. Certain reciprocal relationships between source and absorber are described, and it is shown how in any case in which dosage-rate due to a continuous finite source of radioactive material is known the integral dose due to a point source may be found. It is proved that the integral dose in gm. r throughout any mass due to a milligramme point source for one hour is equivalent to the dosagerate in r/hr. at the position of the point source if the initially absorbing volume be filled uniformly with radiating material of unit density. Several examples of the application of these reciprocal relationships to elements of mass of different shapes such as cylinders, rings, spherical shells, flat discs, etc., are indicated, and a general curve is given showing the integral dose due to a point source near its centre in a mass of approximately spherical shape. Elongation corrections enable estimates to be made for masses of other shapes. Certain other problems are discussed in which absorption in the tissues is taken into account and some reference made to the use of exponential and more general integrals. Tabular data are given for the total energy emitted from point sources in cylindrical filters.Finally, as practical examples the formulæ and methods of the paper are applied to the calculation of integral dose in a head during the use of radium in an antrum. Integral doses due to a lip surface applicator, in an arm bent at right-angles around a lead block as in manipulating radium, and in the trunk as in the treatment of the cervix uteri or the œsophagus, have also been calculated.