Abstract
The use of convolution integrals in the computation of β-ray line shapes is discussed. Line shapes are found by numerical integration for the case of a lens spectrometer in which the only aberrations present are those due to aperture and to size of source. From the computed shapes a relation is found between the half width and the relative size of the two aberrations. The relation is used to estimate the dependence of luminosity on resolution in the case of a prolate spheroidal field spectrometer and of a spectrometer which uses a superposition of this field on a uniform field. The performance is compared with that of a uniform field lens spectrometer of similar size. It seems that each type of instrument may be better than the others in an appropriate range of aperture and of size of source.
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