GOODRICH, Circuit Judge.
This is a civil action based on the “Fair Labor Standards Act of 1938”,1 to recover unpaid overtime wages claimed under § 7 (a)2 and liquidated damages, reasonable attorney’s fee and court costs claimed under § 16(b).3 Defendant is the owner and operator of certain coal mines in Pennsylvania in and around which plaintiffs are employed. That these employees are within the Act is not disputed.
Thirty-eight employees filed complaint seeking recovery as indicated. Defendant coal company in answer to the complaint made application for stay of suit until arbitration had been had. Plaintiffs’ reply to the application was that the contracts set [313]*313out in defendant’s answer and upon which defendant affirmatively relied were illegal and void as against public policy and contrary to the Act. The District Court ordered that trial be “stayed until arbitration has been had in accordance with the terms of the agreements * * * This is the order appealed from. A brief has also been filed for the Administrator of the Wage and Hour Division, United States Department of Labor as amicus curiae. Certain matters of fact were stipulated by the parties,4 which will be considered later in conjunction with the requirements of the Act. Our conclusion is that the matter should proceed to arbitration, but only in accordance with the rulings of law as hereinafter set out.
The events separate naturally into three distinct time periods, each of which centers upon a different problem under the Act. The first period extends from the effective date of the Act, October 24, 1938, until November 1, 1939, when an alleged formula method for overtime was announced. The second period extends from November 1, 1939, until May 1, 1941, when the Union-Operators Agreement of that day went into effect. The third period covers time subsequent to May 1, 1941.
1. First Period, October 24, 1938 to November 1, 1939.
During this first period some of the plaintiffs were paid straight hourly rates and others straight monthly salaries. The defendant coal company admits that neither group of employees were paid any time and one-half overtime wages before November 1, 1939, but contends that the overtime due for the period was later paid by means of an “accord and satisfaction”. This conclusion is disputed by plaintiffs on the ground that whatever was done by employees was a mere “waiver” and ineffective to change their rights under the Act.
Since plaintiffs were employed during the period under contention and since they were not paid the time and one-half rate provided in the Act for weekday hours over forty-four, it is perfectly clear that prima facie there is indicated on behalf of such employees a claim for the pay that they were entitled to under the Act, together with liquidated damages and attorney’s fee. But, says the company, we settled all that controversy by the agreement of May, 1941, made between the union and the company with the union as the authorized bargaining agent for these employee-plaintiffs. We settled it by the provision in the contract that “The [314]*314wage adjustments provided herein for employees customarily working forty-two hours a week, or more, constitute full settlement and satisfaction of all their claims, if any, for overtime against any signatory operator arising out of employment prior to May 1, 1941.”5
What is the effect of this clause in the agreement ? It is now authoritatively settled that a waiver by an employee, even by a release under seal, of his rights against an employer under the Act is not effective to bar him from subsequent assertion of those rights. Brooklyn Savings Bank v. O’Neil, and its companion cases, 65 S.Ct. 895, 900, decide this, as do the flock of well considered cases in the District Courts and Circuit Courts of Appeal.6 But in the Brooklyn Savings Bank decision the Court indicated that other considerations might apply if the sum paid by an employer to an employee was the “result of the settlement of a bona fide dispute * * * with respect to coverage or amount.” The inference is that if there were, between employer and employee, a dispute either as to whether an employee was covered by the Act or, if so. how much he had coming to him by reasons of the provision of the Act, settlement between the two parties of their dispute may, under proper circumstances, be upheld.
But we do not see how the facts of this case, at any rate on the state of the record now before us, bring the claim here involved within the exception mentioned by the Supreme Court as legally permissible. The identity of these employees is established through their lawsuit. We do not understand that the hours or terms upon which they worked prior to May, 1941, is subject to serious dispute. That their employment is covered by the Act is conceded. The amount which in May, 1941, was due them under the Act may be mathematically calculated by simple arithmetic once the hours are separated into regular and overtime hours.7 We have then no settlement of a dispute with respect to coverage or amount involved here. We have, instead, the question of whether a purported settlement can be effected by a modification upward of the wage scale formerly existing. In other words, the company, to meet its obligation incurred by reason of its failure to pay the employees, says that in the future it will pay its men so much. That, it says, may be stipulated as a settlement of claims under the Act which have accrued to the employee. We do not think this is compliance with the Act. The history of the discussions concerning it and the judicial interpretation of the intent of the Act have been developed for us by the Supreme Court and brought into highlight by various dissenting opinions. We think that the change in wage scale is not a settlement for amounts admitted due under the Act prior to the adoption of the new scale and that that case is like the waiver case which the Supreme Court has declared not valid settlement under the statute.
The soundness of this conclusion is demonstrated, we think, by a little further analysis of the considerations which might bear on the differences between a waiver and the settlement under bona fide dispute mentioned by the Supreme Court. The company urges the increased payments under wage adjustments and the advantage of having these adjustments permanently incorporated in the wage scale as the equiva[315]*315lent of the payment which complies with the Act. It must be remembered, however, that the right of time and one-half for overtime, plus liquidated damages for failing to get it, is an individual right of the employee, not a criminal sanction nor even a general right given to the employees’ bargaining unit. Unless an individual employee remained in the company’s employ for long enough after the adoption of this increased wage scale to be paid whatever he had coming under the Act before the wage scale went into effect, he has not been compensated as the Act requires. Furthermore, we do not know, nor see how it could be demonstrated, how much of the increased wage scale was for payment of compensation due under the Act and how much of it may have resulted from economic pressure by the bargaining agent or how much of it came naturally on a rising wage scale market. The contested clause states that the wage adjustments apply to “all claims, if any, for overtime”. We think the very nature of the language used shows that the inclusion of the overtime claims was but part, perhaps only a nominal part, of the bargaining by which the new wage scale was determined. Our conclusion is that the terms of the new wage scale is not a settlement sufficient under the Act in the light of the Supreme Court decisions above cited.
2. Second Period, November 1, 1939, to May 1, 1941.
The second period running from November 1, 1939, to May 1, 1941, involves the interpretation of a resolution of the Board of Conciliation of September 5, 1939, providing:8 “(1) In compliance with the basic hour and rate provision of the Fair Labor Standards Act of 1938, which provides for a maximum of forty hours per week in October, 1940, plus punitive rates for time in excess thereof, it is agreed that the forty hours per week shall be the basis for a formula, which will produce earnings that shall be the same as earnings of such employees for equal time worked;”. It is readily apparent that this clause gives rise to two problems: First, whether the formula described was actually instituted, and, second, whether if instituted it was valid under the Act. If the formula was not actually applied to determine wages then obviously its legality is irrelevant. If it was applied then its legality turns upon precisely the same considerations as the central question of the third period, and would better be considered in conjunction with that question.
The central question of the second period, that is, whether a formula was really adopted, can be answered only by considering what was paid and how it was paid. The language of the clause itself is not at all revealing. The words “forty hours per week shall be the basis for a formula” producing equal earnings for equal time worked indicates nothing beyond an intent expressed on the part of the Conciliation Board to adopt a formula which intent might be imputed to the parties on principles of agency.9 Our problem is, however, not with intent but with realization of intent. As the District Court pointed out, the effect of the resolution and the entire series of agreements which all provided for action by the Arbitration Board, “was to authorize each employer in the industry to adopt any formula which would comply with the Act * * *. There is no doubt that * * * a formula * * * could be worked out.” The coal company used the same reasoning to go one step further and say “Id certum est quod certum reddi potesti.” But it is one thing to say that a formula was authorized, even let us say intended, and quite a bit more to conclude that it was actually applied.
The defendant urges that “the Board of Conciliation by the resolution of September 5, 1939 sought to provide a general formula * * * leaving the mere mathematics of determining each individual rate to bookkeepers.” But the “mere mathematics” of bookkeeping, far from substantiating the view that a formula was in operation, does exactly the opposite. It indicates clearly that however readily a formula might have been made operative none in fact actually was.
Thus we may take a specific example: that of John Sawczak for whom representative due bills are photostatically reproduced and before us. The coal company says: [316]*316“the plaintiff, John Sawczak’s regular rate of pay during the period from October 29, 1939 to October 27, 1940 was $0.60 per hour for the first 42 hours worked in each week, which was the regular hourly rate established for the work done by plaintiff John Sawczak in accordance with the resolution of the Board of Conciliation * * * and one and one-half times that rate for all hours worked by him in excess of 42 hours in any week.”
It will be noted that the base or regular rate was acknowledged to be $0.60 per hour under the resolution of the Board of Conciliation for the first two-thirds of the second period.
Now let us take a due bill and compare. There is reproduced the due bill for November 1-15, 1939. This period comprises two weeks and one day, during which 112 hours were worked and $82.64 earned. If we assume that 8 hours were worked on the extra day then we have a total of 92 hours at regular rate of $0.60 ($55.20) plus 20 hours at time and one-half rate of $0.90 ($18.00) or a total of $73.20. If we assume that the employee did not work at all on the fifteenth day we have, by the same method of computation, a total of $75.60. If we assume that he worked 12 hours on that extra day we only reach a total of $72, or to go still further and assume he worked 16 or 24 hours on that fifteenth day we get respectively $70.80 and $68.40. No computation except one can yield the wages shown on the due bill for the hours worked and that one is under the old rate. When the old rate of $0.7379 shown on the due bills prior to the second period is multiplied by the 112 hours worked we get the precise amount of $82.64 shown as earned and computed to the half cent.
The same procedure could be followed with the other due bills of John Sawczak. Similarly the facts hold for John K. Hohn whose due bills are also photostatically reproduced. And since the parties agree that these samples are indicative of the whole, it is clear what our conclusion must be. The formula under the resolution was never applied. The wages received for the hours worked squared only with the old rates and not with any possible formula rates. The formula may have been easy to apply but the fact is that it was not applied. And the formula can not stand as adopted simply by incantation of the Conciliation Board’s resolution. It can only be shown to have been adopted by utilization. The very ease with which the formula could have been applied — a matter of “mere mathematics” and “bookkeeping” — renders all the more striking the conclusion that it was not done. As to this period it appears that the plaintiffs, at least those whose situation is like that of the employees shown on the samples of due bills introduced by stipulation, can recover for this period without decision of any question of legal validity of any formula. The point is that the old rate was in fact paid and no formula was applied. The effect of the 1941 agreement on the claim has already been disposed of by the discussion under division one of this opinion.
3. Third Period, May 1, 1941, and Following.
The third period covers time subsequent to May 1, 1941. Section 7 of the Wage Agreement of May 20, 1941, gave birth to a new specific formula about which the controversy of the third period turns. The section reads: “(7) The hourly rates of employees, exempted from the provisions of the seven-hour day under the Agreement of May 7, 1936, who have not been adjusted to a forty-hour basis and who regularly worked a fixed number of hours of forty-two hours per week or more, shall be established for the period May 1, 1941 to September 30, 1941, by adding 7%% to their weekly earnings as calculated under the provisions of said agreement and dividing the sum by the hours regularly employed weekly plus one-half the hours worked in excess of forty hours; and for the period October 1, 1941, to April 30, 1943 by adding 10% to their weekly earnings as calculated under the provisions of said agreement and dividing the sum by the hours regularly employed weekly plus one-half the hours worked in excess of forty hours.”
The formula here, unlike that of the second period,10 is specific, detailed and clear. We are confronted with two questions. Was the formula, in fact, applied? Second, if so, is it a valid mode of meeting the requirements of the Act?
[317]*317The wage increases were clearly granted. Wage increases, however, were not required by the Act, but left to the bargain between the parties. What the statute requires is a fixed rate for the regular hours and a time and one-half rate for overtime hours. But for this period, as for the second period, we find that the old rates and not the formula rates fixed the amount of the employee’s pay.
It is not necessary to repeat similar arithmetical calculations.11 Suffice it to add that of the four sample due bills submitted for John Sawczak for this third period, one actually carries the rate printed on it as “.8116”. Now if the number of hours on this due bill, 120, are divided into the earnings, $97.36, we get a straight-rate quotient of $.8113, a difference of only 3/100 of a cent. It is obvious that the actual rate used was a straight-time rate and not the formula rate. Employees’ wages were figured just as before. There was no application of the regular rate and overtime rate at all. Instead, there was complete reliance upon the fact that a formula had been determined and announced in writing in a wage agreement. But in the one place where an announcement of rates actually used might really be determinative, i.e., on the due bills, there was not merely silence as to regular and overtime rates but obvious evidence that the straight rate method had been used. It is true that on this very due bill, there also appeared the following: “the earnings shown on this due bill have been computed in accordance with the provisions of section (7) of agreement of May 20, 1941.” But, of course, the law does not recognize the mere recital of an act as done, as the equivalent of the actual doing of the act.
Even the method of taking deductions for absences was premised on a straight-rate system rather than the required method. Thus the May, 1941 agreement provides : “To avoid undue deductions for absences the parties agree that the rate of deduction per day from weekly earnings for absence of any such employee shall be determined by dividing his weekly earnings, when regularly employed, by the number of days in his regular work week.” On a straight-rate basis it will be seen the total earnings divided by days worked gives a perfectly fair figure per day. But on a regular plus overtime basis, the particular day in the week that is missed is the chief factor since it may be an overtime day rather than a regular rate day.
So for this period, as well as the second period, the conclusion is clear that formula and practice did not follow each other. The employees continued to be paid on a straight time basis and not by a giving of the increased amount for overtime provided in the statute.
The Formula
Consideration of the formula is the next step, lest it be concluded that a change in method of computation on a due bill would show compliance with the Act.
The Supreme Court in a series of cases extending from June, 1942, until June, 1945, has laid down the outlines within which a formula must fall to meet the statutory requirements. The crux of the matter is that the regular rate, however established, must be a bona fide rate and not a fictitious one. In Overnight Motor Transportation Co., Inc., v. Missel, 1942, 316 U.S. 572, 62 S.Ct. 1216, 86 L.Ed. 1682, which together with Walling v. A. H. Belo Corp., 1942, 316 U.S. 624, 62 S.Ct. 1223, 86 L.Ed. 1716, decided on the same day, constitutes the first sharp analysis by the Supreme Court of the question, the court laid down the principle that enabled a regular rate to [318]*318be determined when it was not specifically stated. The court there said [316 U.S. 572, 62 S.Ct. 1221]: “Neither the wage, the hour nor the overtime provisions of sections 6 and 7 on their passage spoke specifically of any other method of paying wages except by hourly rate. But we have no doubt that pay by the week, to be reduced by some method of computation to hourly rates, was also covered by the act.” To determine the regular hourly rate thus became a mere matter of division under the Missel case rationale. See Walling v. Helmerich & Payne, Inc., 1944, 323 U.S. 37, 40, 65 S.Ct. 11.
But whether the regular rate was explicitly stated or had to be figured out as indicated, the essential test of its sufficiency was whether it was an actual regular rate or a fictitious one. The defendant here, as defendants have in other cases under the Act, relies heavily on the Belo decision, supra. In that case the C6urt upheld a contract rate as a legal regular rate. Belo is readily distinguished from the instant case. There only the overtime varied, though never below the 150% minimum provided by law. Thus the Court said [316 U.S. 624, 62 S.Ct. 1227]: “To be sure, $1,753 is more than 150% of $.67. But the Act does not prohibit paying more; it requires only that the overtime rate be ‘not less than’ 150% of the basic rate. It is also true that under this formula the overtime rate per hour may vary from week to week. But nothing in the Act forbids such fluctuation.” Furthermore, in the Belo situation the expressed purpose was not to keep the pay levels exactly the same as before the Act (which would at least put one on inquiry as to whether the so-called regular rate was fictitious) but only to do so insofar as possible under the Act.
In the instant case we have neither the limitation of purpose to make the new wage payment plan a duplicate of the old insofar as possible nor the limitation of fluctuation only to the overtime rate. Both regular and overtime fluctuate in our formulae. Both these limitations apply to the Belo case and serve to distinguish it on principle. Furthermore, in the words used by the Supreme Court to distinguish the Belo case from Walling v. Youngerman-Reynolds Co., 65 S.Ct. 1242, 1246, 1250: “This Court’s decision in Walling v. A. H. Belo Corp., 316 U.S. 624, 62 S.Ct. 1223, 86 L.Ed. 1716, lends no support to respondent’s position. The particular wage agreements there involved were upheld because it was felt that in fixing a rate of 67 cents an hour the contracts did in fact set the actual regular rate at which the workers were employed. The case is no authority, however, for the proposition that the regular rate may be fixed by contract at a point completely unrelated to the payments actually and normally received each week by the employees.”
Following the ground clearing decisions of the Missel and Belo cases came three Supreme Court decisions which we think directly applicable to the instant case. In Walling v. Helmerich & Payne, Inc., 1944, 323 U.S. 37, 65 S.Ct. 11, 12, the attempt to maintain wage levels as before was by means of contracts under the so-called Pox-on or split-day plan. The actual mechanics of the Poxon formula were that each tour was arbitrarily split in half and the first part compensated by regular wage rates while the second part received the overtime rate of 150%. There was thus an apparent compliance with the Act. But actually because of the figures at which the new regular rate was set, exactly the same wages as before were paid. This is precisely what has occurred under the formula of the instant case. The Court’s analysis of the workings of the Poxon plan shows to what full degree it is the essence of the formulae of the instant case: “These so-called ‘regular’ and ‘overtime’ hourly rates were calculated so as to insure that the total wages for each tour would continue the same as under the original contracts, thereby avoiding the necessity of increasing wages or decreasing hours of work as the statutory maximum workweek of 40 hours became effective. Only in the extremely unlikely case where an employee’s tours [of duty] totalled more than 80 hours in a week did he become entitled to any pay in addition to the regular tour wages that he would have received prior to the adoption of the split-day plan. Until more than 80 hours had been worked the plan operated so that the employee could not be credited with more than 40 hours of ‘regular’ work, the remaining time being denominated ‘overtime.’ Hence, since the wages under the old system and under the split-day plan were identical, the original tour rates could be used as the simple method of computing wages for each pay period. The actual and regular workweek was accordingly shorn of all significance'.” And the conclusions of the Court in the Helm [319]*319erich case apply equally well to our case: “The vice of respondent’s plan lay in the fact that the- contract regular rate did not represent the rate which was actually paid for ordinary, non-overtime hours, nor did it allow extra compensation to be paid for true overtime hours. It was derived not from the actual hours and wages but from ingenious mathematical manipulations, with the sole purpose being to perpetuate the pre-statutory wage scale.” “Any other conclusion in this case would exalt ingenuity over reality and would open the door to insidious disregard of the rights protected by the Act.”
Any formula, whatever its type, whether Poxon plan or a plan similar to that used in the case at bar or some other plan, so long as it is conditioned entirely upon maintaining actually and essentially the same mode of payment as before the Act, is inevitably illegal in that it creates a fictitious regular rate. In Walling v. Youngerman-Reynolds Hardware Co., 65 S.Ct. 1242, 1250, a formula was devised which although conditioned upon maintaining the old mode of payment unaltered was patterned in general terms upon the Belo formula’s guaranty requirement in an attempt to run the gantlet of the Act. But the very guaranty feature was so devised that under it employees would be paid by a piece rate as before. The plan failed to meet the statutory requirement.
In Walling v. Harnischfeger Corp., 65 S.Ct. 1246, 1248 (decided the same day as the Youngerman-Reynolds case, supra, June 4, 1945) the Court further added: “Our attention here is focused upon a determination of the regular rate of compensation at which the incentive workers are employed. To discover that rate, as in the Youngerman-Reynolds case, we look not to contract nomenclature but to the actual payments, exclusive of those paid for overtime, which the parties have agreed shall be paid during each workweek.”
From what has been said in this review of Supreme Court opinion it is clear that for a formula to be productive of a regular rate and overtime rate such as will meet the requirements of the Act, it must not be a mere mathematical cloak under which to continue the same payment practices as hitherto. The rate established as the regular rate, in order to be adjudged actual and not fictitious, must either (1) bear a real relation to actual earnings as deduced by the Missel case principle or (2) it must be a genuine contractual decrease in wages, based upon bargaining individual or collective. Under such a test of the requirements of the Act, the formulae of the instant case clearly fail. The evidence already marshalled is clearly indicative that the formulae were directed only to one goal, i.e., to continue the old methods of payment for work while ostensibly meeting the requirements of the Act technically and technicalities are not enough.
Arbitration
Our final problem has to do with the question of arbitration. As stated above the District Court stayed the lawsuit until arbitration could be had. The provision in the contract of the parties for arbitration goes clear back to the year 1903 following the settlement in the famous anthracite coal strike of that period.12 There are three points with regard to the arbitration here.
[320]*320The first point; has to do with the question whether the formula and the waiver provisions which we have found to be insufficient under the Act so completely vitiate the contract for illegality that no reference to arbitration can be made. We think this question must be answered in the negative. The sufficiency of the wage formula and the provision for waiver are entirely separable elements of the contract between the parties. We do not refer to arbitration the question of legality of the formula. That is a question of law which the Court must take responsibility in answering. All we are saying upon this point is that the arbitration provision is not rendered ineffective because the contract contains one clause, setting out the formula, and another clause setting forth a provision for waiver which we deem insufficient under the statute.
The next point is whether there is anything to refer to arbitration. We think the answer to this question is yes. After the rights of claimants have been determined as a matter of law there is still the not inconsiderable problem of determining how much each claimant is entitled to. That involves his identity as an employee of the defendant, the question of the days and hours on which he worked and the calculation of the amount of money to which he may be entitled under the Act.13 It is the kind of problem which is properly referable either to a special master or an arbitrator, and since the parties by contract have provided for arbitration we think the reference is appropriate in this instance.
The third question has to do with the application to this situation of the United States Arbitration Act, 9 U.S.C.A. § 1 et seq. We have already answered that question in Donahue v. Susquehanna Collieries Co., 1943, 138 F.2d 3, 149 A.L.R. 271, and the District Court followed that decision in making the order which it did. Our analysis of the problem involved has found support since in a decision by, the Fourth Circuit in Agostini Bros. Bldg. Corp. v. United States, 1944, 142 F.2d 854. The Sixth Circuit has doubted the correctness of our result. Gatliff Coal Co. v. Cox, 1944, 142 F.2d 876.14
Without repeating the discussion of the problem which we made in the Donahue decision, we adhere to the result there reached and for the reasons there given. We think the criticism made by our colleagues in the Sixth Circuit is not convincing. While it is true that in defining commerce in section 1 contracts of employment of specified types of employees were excluded, it should be noted that this exclusion was only in the definition of commerce in that section. Section 3 of the statute15 does not use the [321]*321term commerce. We do not think that the limitation in the definition in section 1 should be applied as an over-all limitation elsewhere to the section where the defined term is not used. We, therefore, adhere to the Donahue decision and with increased confidence because of its emphatic approval by the Fourth Circuit in the case just cited.
The stay order, in the District Court, by which the action is stayed pending arbitration, should be modified so that the proceedings in arbitration will be conducted under the rules of law as set out herein. As thus modified the order of the Court below is affirmed.