VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. 475 



alkali is added, this mixture submitted to distillation, with the addition of con- 

 centrated vitriolic acid, yields no red vapours, and very little smell of nitrous 

 acid. 



And to Mr. Keir's letter on the analysis of the same acid, he adds, if, on 

 examining the acids which you or others may hereafter obtain by the inflamma- 

 tion of airs, a mixture of marine acid be constantly found to accompany the 

 production of the nitrous, the fact will be only analogous to all the other known 

 productions of nitrous acid; in all which, either in the natural formation of 

 nitre as in Spain and India, or in the nitre beds and walls made by art, a very 

 large proportion of marine salts is constantly observed to accompany the nitre. 

 Other particulars on this subject may be found by consulting the pamphlet in 

 which this paper was separately printed by Dr. Priestley. 



XX. On the Probabilities of Survivorships between Two Persons of any Given 

 ^ges, and the Method of Determining the Falues of Reversions Depending on 

 those Survivorships, By Mr. Win. Morgan, p. 331. 



The hypothesis of an equal decrement of life, adopted by M. de Moivre, for 

 the purpose of facilitating the computations of life annuities, has not only been 

 rendered unnecessary by the late publication of many excellent tables deduced 

 from real observations, but has also been found so very incorrect in some cases, 

 that probably little or no recourse will ever be had to it in future. But though 

 the direct application of this hypothesis may be laid aside, there is danger of its 

 not being entirely abandoned ; and mathematicians may still be led to reason 

 from this principle, by deriving their rules from the expectations rather than 

 from the real probabilities of life. The ingenious Mr. Thomas Simpson has 

 contented himself with this inaccurate method in his select exercises, and he 

 has been followed in it by most other writers on the subject. Even in those 

 cases which involve only 2 lives, the errors are often considerable, especially if 

 the expectations are derived from the London, the Sweden, or any other tables 

 in which the decrements of life are unequal. But when 3 lives are involved in 

 the question, these errors are generally enormous; nor is it ever safe, when the 

 ages of those lives differ very much, to have recourse to rules which are founded 

 on this principle. The 3 following problems, though the most common in the 

 doctrine of survivorships, have never hitherto been solved in a manner strictly 

 true. The 2d of them is of particular importance, and I have taken much 

 pains to examine how far Mr. Simpson's solution of it may be depended on. It 

 has indeed been solved by M. de Moivre, and Mr. Dodson: but the first of these 

 writers has erred most egregiously in the solution itself, and the other having 

 derived his rule from a wrong hypothesis, has rendered it of no use. It is much 

 to be wished that the solutions of all cases in reversions and survivorships were 



3 p 2 



