The transaction approach to measuring net income is the traditional bookkeeping and accounting method. That is, individual transactions such as each sale, each purchase, and every expense are recorded into general ledger accounts. At any point you can go to an account such as Salaries Expense for Sales Staff and see the year to date amount of such an expense. With the use of accounting software, an enormous quantity of transactions can be recorded into many detailed accounts.
I believe that the balance sheet approach is also referred to as the capital maintenance approach. Under the balance sheet approach one looks at the change in stockholders' or owner's equity to determine the amount of net income during the period between balance sheets. This approach requires that you exclude any additional capital from the owners as well as any dividends or withdrawals distributed to the owners. For example, if stockholders' equity increased by $5 million with $2 million caused by the issuance of new shares of stock, and $1 million distributed as dividends, the net income would have been $4 million. We can verify the calculation with the following: net income of $4 (an addition to equity) plus new investor money of $2 (an addition to equity) = $6 of additions to equity, minus dividends of $1 (a decrease to equity) = $5 (the net increase to equity). Under this balance sheet approach you will not have the detailed information on revenues and expenses that would be available under the transaction approach.