The purpose of this article is to call attention to certain fundamental relationships among those basic elements which constitute the distinguishing features of accounting problems involving the admission of a new partner by investment. These relationships are presented against the background of a simple illustrative problem of which the following is a preliminary statement. To explain the case a situation of a partnership firm is cited, in which a new partner is consented to admission by other partners at a pre-assumed equity ratio. The situation that rose due to the inclusion of another partner allows for the emergence of any one of three possible situations. The first presents a condition of no bonus, no goodwill. In other words the situation is termed as "par basis." The second situation entitles the company to a bonus or goodwill. The third situation entitles the new partner to a bonus or goodwill. This article analyzes each situation and compares their results. The investigation on which this article is based produced a number of expressions and corresponding definitions relating to the dollar values of bonus and goodwill.