In this paper an approximate confidence interval is found for the expected value of the difference between two quantities which are independently distributed proportionally to $\chi^2$ variates. Three methods are used. The first is based on the work of Welch [13], [14] and Aspin [1], [2] on the generalized "Student's" problem, and involves neglecting successively higher powers of the reciprocal of one of the degrees of freedom. This method is used to check the other two solutions, both of which involve neglecting successive increasing and decreasing powers, respectively, of a nuisance parameter. Finally a solution is formed using those resulting from the second and third methods, and is more accurate than those solutions. The order of accuracy, and the use of the final solution, are discussed. The paper does not present a method of computing confidence intervals in a form suitable for immediate practical application. Series developments of a certain hypothetical function are given; more remains to be said about the relation between the series and the function, and the problem of computing tables. A computational exploration of the solution is at present in hand.