In a paper read to the Society on March 1 (ante, p. 310) I spoke of the application of some of its results to the solution of equations involving an unknown Linear and Vector Function. These results depended chiefly upon the expression of the function in terms of its roots, scalar and directional; and I now give a few instances of their utility, keeping in view rather variety of treatment than complexity of subject. The matter admits of practically infinite development, even when we keep to very simple forms of equation, and is thus specially qualified to show the richness in resources which is so characteristic of quaternions. But it will be seen also to be strongly suggestive of the extreme caution required even in the most elementary parts of this field of inquiry.