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747IjiriTo comprehend what this extensionmeans in the accounting environment, itis useful to translate stock and flow accounts into accounting terms, "wealth"and "income." Wealth is assets less liabilities. Income is revenues less expenses.There are other flow accounts such ascontributions by owners or distributionsto owners, as well as such flow accountsas purchases, collections, or payments,which do not normally give rise to achange in wealth. But since income-producing activities are the primary sourceof wealth change of interest, flow accounts other than income will be setaside for simplicity of illustration.MOMENTUM,FORCE,AND IMPULSETo extend the wealth-income dichotomy to a trichotomy, it is necessary tointroduce the concept of "the rate atwhich wealth is changing," or equivalently "the rate at which income is beingearned." The concept will be denoted bymomentum. While both wealth and income are measured in monetary unitssuch as dollars, momentum is measuredin monetary units per period, such asdollars per year or per month. Unlike income, for which measurement requiresspecifying two points in time, momentum shows a state (of income beingearned) at any single point in time.Income is then a realization of momentum as time passes.4 Under a stablemomentum, income is defined as momentum times "duration" of momentum.5The concept of momentum is actuallyused in business, although normally thedifference between momentum andincome is not explicitly made. For example, when an executive says "this investment (or that division or that asset) isearning 60 million a year," what theexecutive refers to is momentum, notincome.6Because momentum may be definedand measured at any single point in time,it is conceivable to prepare a "balancesheet" in which all revenues and expenses are treated as stock accounts andmeasured in momentum. Such measurements may be based on a detailedanalysis of asset and liability items interms of their ability to contribute tothe earning process. In this analysis,momentum for monetary assets and liabilities is easy to determine using thepertinent interest rate. For nonmonetaryitems, an analogy with the historical costprinciple may be taken, and we mayadopt what may be called the "historicalmomentum principle." Under this principle, the momentum of an itemacquired is set equal to the momentumof an item given up in exchange, until thechange in momentum is confirmed byrealization, such as actual sale or lease orelse a contract to sell or lease at a fixedprice or rate.adding to - 10,000, explain the reason. The next stepwill be to treat the loss of 10,000 as an explanundum andtry to find its explanans.4 If wealth at time Tis denoted by WT, income betweentime T' and T is WT- Wt., while momentum is d W/dtevaluated at t T.5 An investment of 10,000 yielding interest at the rateof one percent per month has a momentum of 100/mo.If the momentum is stable at this rate, its realization overa year (a one-year duration) will result in an income of 1,200. In mechanics, momentum is defined as mass (m)times velocity (v). Analogously, in the accountinginterpretation, momentum may be defined as investmenttimes yield, e.g. 100/mo 10,000 x I W6o/mo.6 A car driven at 60 miles an hour at a given point intime may or may not go 60 miles in the next hour since thespeed need not remain constant.I The historical momentum principle implies appreciation or depreciation in book value at a constant rate,hence it conflicts with the historical cost principle whichresults in a constant book value. A translation of suchmethods as FIFO, LIFO, the lower-of-cost-or-marketmethod, and various depreciation methods also generates a different result when applied to momentumcompared with their application to wealth. Comparisonsand reconciliations of the differences between accounting principles and standards as they are applied to wealthand as they are applied to momentum offer an interest-This content downloaded from 195.78.109.162 on Sun, 15 Jun 2014 12:59:38 PMAll use subject to JSTOR Terms and Conditions

The Accounting Review, October 1986748Consider the fact that momentumitself changes over time. Just as the rateof change in wealth is measured inmomentum, the rate of change in momentum must now be measured inanother concept.8 Because there is nosuch concept currently established in accounting, we may seek a concept inmechanics that fits in the definition.That is the concept of force. Force ismeasured as a rate of change in momentum. Hence, in the accounting interpretation, it is expressed in monetary unitsper period squared, such as dollars permonth per month.9Although both momentum and forceare undoubtedly terms that are notfamiliar to accountants and managers,such an introduction of new terms isunavoidable in an extension of a systemlike double-entry bookkeeping. Fortunately, both terms have common-senseinterpretations that fit very well with thetechnical meaning we are seeking."?Since momentum shows a state at agiven point in time, equation (1) may beapplied to momentum, treating it asstock. Namely, we want to account forchanges in the net balance of momentumby means of a set of accounts designed toexpress the reasons for the change inmomentum. Here again, since there is noconvenient term that currently exists inaccounting to describe a set of such accounts, a comparable term in mechanicsmay be considered.There are two requirements that thenew concept must satisfy. Just as wealthand income are both in the same measurement unit, for example, dollars,momentum and this new concept mustbe in the same measurement unit, forexample, dollars per month. Second,just as income is a realization of momentum and is measured by momentumtimes duration when the momentum isstable, the new concept is a realization offorce and is measured by force timesduration when the force is stable. A concept in mechanics that satisfies bothrequirements is called impulse.IFORA FRAMEWORKTRIPLE-ENTRY BOOKKEEPINGTable 1 shows the relationship amongwealth, momentum, income, force, iming subject of study, enhancing our understanding ofwhat these accounting principles and standards meanin a broader context.a"Why did our momentum go down from 5 million amonth to 4 million a month?" an executive may ask;"What are the contributing factors?" Often casualexplanations are given in monthly management reportsto the board of directors. We need to formalize the reasoning behind the attribution to various factors in theform of accounts.9 In mechanics, force is defined as mass (m) timesacceleration (a). In the accounting interpretation, forcemay be defined as investment times acceleration (rate ofchange in yield), e.g., 10/mo2 10,000x0.1 %/mo2,namely the yield, stated as a fraction per month, increases by 0.1 Wo/mo for each one-month duration of,say, an inflationary force. If wealth is W, momentum isM, and force is F, then M dW/dt, and F dM/dt d2W/dt2. Here, the measurement of wealth is theprimary measurement and the measurement of momentum and that of force are derived measurements. It hasbeen observed, however, that the "best" measurementfor one of the three need not be the "best" measurementfor the remaining two. See Ijiri and Noel [1984] for adetailed discussion on the choice of measurement foreach of the three dimensions, comparing the historicalcost and the current cost measurements.'? Concepts and relationships among them in anoriginal field must of course fit in nicely in the appliedfield; but it is also nice if the concepts and relationshipsare not inconsistent with their common-sense interpretations.11In mechanics, impulse is measured as force timesduration. It can also be shown that it is equivalent tomass times velocity, just as momentum is. Although inmechanics impulse is used in particular for situationswhere duration is short, in our accounting application wemay apply it more generally, irrespective of the length ofduration. For a detailed treatment of the concepts andmeasurements of momentum, force, and impulse inNewtonian mechanics, see, for example, French [19711.The analogy with Newtonian mechanics should, however, be pursued only to obtain some valuable insightsinto how the triple-entry bookkeeping framework shouldbe developed. A wholesale translation without regard toits applicability to accounting is dangerous, since afterall, physics and accounting are two different fields ofendeavor.This content downloaded from 195.78.109.162 on Sun, 15 Jun 2014 12:59:38 PMAll use subject to JSTOR Terms and Conditions

Ijiri749TABLEIA eAccountingSingle-EntryBookkeepingin EntryMomentumBookeepingin tementActionin DollarsForceStatement-P-ADerivativeRelationship--- VAn IntegralRelationship-- A DifferenceRelationshippulse, and one more concept yet to beexplained,namelyaction. On the bottomrow of Table 1 we have the familiartwodimensionsin wealthaccounting,that is,wealth and income, setting aside thethird dimensionof action for now. Thetwo dimensionsare tied togetherby thedifference relationship, AWealth Income, both of which are stated in monetary units such as dollars.We now move to the second row ofTable 1, which expresses the doubleentry equation (1) applied to momentum, namely AMomentum Impulse.That is, changes in momentum at anytwo points in time are fully accountedfor by impulse, whose accounts expressthe underlyingforces that are judged tobe responsible for the momentumchange. Both momentum and impulseare stated in monetaryunits per period.This particular layer of accounting,called momentum accounting, is analogous to the conventional double-entryframework for wealth and income, except that the primary object of accounting is momentum instead of wealth.The top row of Table 1 shows the dimension for force accounting. Its onlydimension is force, which is stated inmonetary units per period squared.While conceptually it is possible to define a concept that deals with a rate ofchange in force, a higher dimension isnot considered at this time. (Such aconcept will become necessary if thebookkeeping system needs to be furtherextended to quadruple-entry bookkeeping.) Hence, force accounting has to bebased on a single-entry bookkeeping system.Two types of relationships exist acrossThis content downloaded from 195.78.109.162 on Sun, 15 Jun 2014 12:59:38 PMAll use subject to JSTOR Terms and Conditions

750The Accounting Review, October 1986different layers of accounting. One is thederivative relationship that exists between wealth and momentum as well asbetween momentum and force. One dealswith the rate of change in the other, andmathematically one is the time derivativeof the other. This relationship is shownin Table 1 by diagonal arrows. The otheris an integral realtionship connectingconcepts vertically in Table 1, such asmomentum-income and force-impulse(as well as impulse-action, which will beexplained later). For a constant momentum or force, income is momentumtimes duration and impulse is force timesduration, as mentioned earlier. Moregenerally, income is measured as anintegration of momentum, and impulseis measured as an integration of force,both over a given period of time.The diagonal, derivative relationshipand the vertical, integral relationshipthat tie wealth, momentum, and incomeare the converse of each other if the integration constant is set equal to the beginning balance of wealth. Hence, theequality holds between the wealth dimension and the income dimension whenthe beginn

746 The Accounting Review, October 1986 under which some of the fundamental concepts in management accounting, es- pecially those in variance analysis, can be extended and integrated with those of financial accounting. WEALTH AND INCOME The key to an extension of any system lies in understandi