It is with the independent formula (8.) that we have to do. Therefore it must be remembered that the conditions for the value of are now modified, and that is a perfectly arbitrary whole number. This circumstance, combined with the fact (which we may easily perceive) that whatever is, every term of (8.) after the ()th is = 0, and that the ()th term itself is always enables us to find the value (either numerical or algebraical) of any th Number of Bernoulli , in terms of all the preceding ones, if we but know the values of , ... . We append to this Note a Diagram and Table, containing the details of the computation for , (, , being supposed given). On attentively considering (8.), we shall likewise perceive that we may derive from it the numerical value of every Number of Bernoulli in succession, from the very beginning, ad infinitum, by the following series of computations:— 1st Series.—Let , and calculate (8.) for this value of . The result is . 2nd Series.—Let . Calculate (8.) for this value of substituting the value of , just obtained. The result is . 3rd Series.—Let . Calculate (8.) for this value of , substituting the values of , before obtained. The result is . And so on, to any extent. The diagram[30] represents the columns of the engine when just [Pg 61] prepared for computing , (in the case of ); while the table beneath them presents a complete simultaneous view of all the successive changes which these columns then severally pass through in order to perform the computation. (The reader is referred to Note D, for explanations respecting the nature and notation of such tables.)