California levies a tax on the net income of corporations engaged exclusively in interstate commerce. Luckenbach Steamship Company, concededly subject to the tax, takes issue with the formula employed by the state Franchise Tax Board in allocating to California a portion of its net corporate income from interstate operations during the years 1942 through 1947.

Luckenbach is a Delaware corporation with its principal place of business in New York City. At the outbreak of World War II it was operating a fleet of merchant vessels as a common carrier, principally in intercoastal trade, with stops at California ports. In 1942 wartime exigencies caused the federal government to requisition the ships of Luckenbach and other carriers and to place them under a “time charter” arrangement administered by the War Shipping Administration. Under this arrangement, which prevailed during most of 1942 and 1943, the government controlled the vessels ’ movements but Luckenbach continued as operator of the ships. Effective about the beginning of 1944 the government requisitioned the vessels on a “bareboat charter” basis under which the War Shipping Administration became the actual operator. Luckenbach retained legal title and acted as general agent for these and other government-operated vessels. During 1946, the vessels (other than several which were lost) were returned to Luckenbach, which resumed intercoastal common carrier operations.

Luckenbach filed California income tax returns for the years 1942 through 1947, utilizing a so-called “voyage-day” formula by which to compute that portion of its corporate income attributable, in its view, to California sources. Dissatisfied with the formula used by Luckenbach, the Franchise Tax Board levied an assessment for larger amounts, using the so-called “port-day” formula in the computation of income attributable to California. Luckenbach paid the additional taxes under protest and sued to recover. The trial court denied recovery and Luckenbach appeals. Sole issue on appeal is lawfulness of the port-day formula employed by the taxing agency as applied to income derived from operation of vessels during the years 1942 through 1947.

The problem is a perennial one in state taxation of multistate business concerns. Where a firm’s business operations within and outside the taxing state are so closely integrated that each is dependent upon and contributes to the other, its income is treated as a unit, and it is regarded as a “unitary business” for state tax purposes. (Butler Bros. v. McColgan, 315 U.S. 501 [ 62 S.Ct. 701 , 86 L.Ed. 991 ], affirming 17 Cal.2d 664 [ 111 P.2d 334 ] ; Bass, Ratcliff & Gretton, Ltd. v. State Tax Com., 266 U.S. 271 [ 45 S.Ct. 82 , 69 L.Ed. 282 ]; Pacific Fruit Express Co. v. McColgan, 67 Cal.App.2d 93 [ 153 P.2d 607 ].) Separate accounting of in-state activities of a unitary business does not accomplish clear segregation of net income attributable to sources within the taxing state. (John Deere Plow Co. v. Franchise Tax Board, 38 Cal.2d 214, 223 [ 238 P.2d 569 ].) In order to allocate a part of the net income subject to state income taxation, the taxing state may apply a formula fairly calculated to reflect the relative contribution of in-state activities to total net income. (El Dorado Oil Works v. McColgan, 34 Cal.2d 731, 738 [ 215 P.2d 4 ]; Edison California Stores, Inc. v. McColgan, 30 Cal.2d 472, 479-480 [ 183 P.2d 16 ] ; Keesling and Warren, The Unitary Concept in the Allocation of Income, 12 Hastings L.J. 42 ; Altman and Keesling, Allocation of Income in State Taxation (2d ed. 1950) p. 107.) Various states have adopted varying formulae. (Hartman, State Taxation of Corporate Income from a Multistate Business, 13 Vand.L.Rev. 21, 65.)

California, like a number of other states, employs a three-factor formula consisting of property, payroll and revenue. A simplified statement of the formula is that the average of the percentages of property, payroll and revenue located or occurring in California is applied to unitary net income, in order to establish that portion attributable to California. Validity of the three-factor formula as an abstract proposition has been judicially confirmed and is not challenged here. (Butler Bros. v. McColgan, supra, 315 U.S. at p. 509 [ 62 S.Ct. 701 , 86 L.Ed. at p. 997 ]; El Dorado Oil Works v. McColgan, supra, 34 Cal.2d at p. 738 ; Edison California Stores, Inc. v. McColgan, supra, 30 Cal.2d at p. 479 .) What is challenged is the subordinate port-day formula used by the taxing agency as an adjunct of the property-payroll-revenue formula in computing Luekenbach’s taxable income from oceangoing vessels. The ratio of the total days spent by a vessel in California ports during a given year to its total days spent in all ports was calculated. That ratio was applied to the vessel’s value, payroll and revenue, as a preliminary to application of the standard three-factor formula. The port-day method is expressed fractionally as follows;

Number of days in California ports

Number of days in all ports

The voyage-day formula urged by Luckenbach apportions the vessel’s income to California on the basis of ratio of the number of voyage days (including days in port) in California to the total number of voyage days during the period. Fractionally, this is expressed:

Number of days in California

Number of days in and out of California

Thus, in the case of a ship which occupied 10 per cent of its time in California ports, 30 per cent in other ports and 60 per cent on the high seas, the California allocation of property-payroll-revenue would be one-quarter under the port-day formula but only 10 per cent under the voyage-day method.

Luckenbach points out that vessels earn income not only in ports but while traveling the high seas, outside the territorial limits of California or any other taxing jurisdiction; that a fair apportionment formula would allocate some portion of vessel income to the location (high seas) where it is earned; that the port-day formula refuses recognition to vessel time on the high seas, results in extraterritorial taxation of income earned on the high seas and violates the due process and interstate commerce provisions of the federal Constitution, as well as the then current provisions of the taxing law itself. The argument emphasizes location of the income-producing activity as the dominant element in apportioning unitary income. (See Hartman, op. cit. supra, 13 Vand.L.Rev. at pp. 663-665.)

The courts have had numerous occasions to consider local ad valorem taxability of vessels plying the high seas in interstate and foreign commerce. (See cases cited in Scandinavian Airlines System, Inc. v. County of Los Angeles, 56 Cal.2d 11, 21-24, 45 [ 14 Cal.Rptr. 25 , 363 P.2d 25 ].) So far as we are aware, this is the first case considering apportionment of unitary income derived exclusively from ocean-going vessels.

The due process phase of the taxpayer’s attack rests on the general principle that states may not tax income of foreign corporations derived from extraterritorial sources. (Connecticut General Life Ins. Co. v. Johnson, 303 U.S. 77 [ 58 S.Ct. 436 , 82 L.Ed. 673 ]; James v. Dravo Contracting Co., 302 U.S. 134 [ 58 S.Ct. 208 , 82 L.Ed. 155 , 114 A.L.R. 318 ]; Hans Rees’ Sons, Inc. v. North Carolina, 283 U.S. 123 [ 51 S.Ct. 385 , 75 L.Ed. 879 ].) The limit of taxability has been described in various ways. In assessing taxes against interstate income, the state may not tax “extraterritorial values.” (Butler Bros. v. McColgan, supra, 315 U.S. at p. 507 [ 62 S.Ct. 701 , 86 L.Ed. at p. 996 ].) Its taxing power must bear “fiscal relation to protection, opportunities and benefits given by the state”; a controlling question is “whether the state has given anything for which it can ask return.” (Wisconsin v. J. C. Penney Co., 311 U.S. 435, 444 [ 61 S.Ct. 246 , 85 L.Ed. 267, 270-271 ].) A formula for apportionment of unitary income must be “fairly calculated” to assign to the taxing state that portion of income “reasonably attributable” to the business done there. (Butler Bros. v. McColgan, supra, 315 U.S. at p. 506 [ 62 S.Ct. 701 , 86 L.Ed at p. 995]; Bass, Ratcliff & Gretton, Ltd. v. State Tax Com., supra, 266 U.S. at pp. 281-282 [ 45 S.Ct. 82 , 69 L.Ed. at pp. 286-287]; Southern Pacific Co. v. McColgan, 68 Cal.App.2d 48, 60 [ 156 P.2d 81 ].) A formula valid in its general application must pass the test of fairness as applied to the individual taxpayer’s situation. (Norfolk & Western Ry. Co. v. North Carolina, 297 U.S. 682, 685 [ 56 S.Ct. 625 , 80 L.Ed. 977, 980 ]; Pacific Fruit Express Co. v. McColgan, supra, 67 Cal.App.2d at pp. 99-100.) If the formula attributes to the taxing state income “out of all appropriate proportion” to the business done there, the formula and the tax will fall. (Hans Rees’ Sons, Inc. v. North Carolina, supra, 283 U.S. at p. 135 [ 51 S.Ct. 385 , 75 L.Ed. at pp. 907-908].) One who attacks a formula of apportionment carries a “distinct burden” of showing by “clear and cogent evidence” that extraterritorial values are being taxed. (Butler Bros. v. McColgan, supra, 315 U.S. at p. 507 [ 62 S.Ct. 701 , 86 L.Ed. at p. 996 ] ; Norfolk & Western Ry. Co. v. North Carolina, supra, 297 U.S. at p. 688 [ 56 S.Ct. 625 , 80 L.Ed. at p. 982 ]; El Dorado Oil Works v. McColgan, supra, 34 Cal.2d at p. 744 .)

It is impossible to divide the revenues and costs of a unitary business in accordance with state lines. (Norfolk & Western Ry. Co. v. North Carolina, supra, 297 U.S. at p. 684 [ 56 S.Ct. 625 , 80 L.Ed. at p. 979]; Underwood Typewriter Co. v. Chamberlain, 254 U.S. 113, 121 [ 41 S.Ct. 45 , 65 L.Ed. 165, 169-170 ].) Recognizing the artificiality of formulary devices, the courts have demonstrated a readiness to sustain any formula reasonably designed to determine the net income attributable to business done in the taxing state. (Northwestern States Portland Cement Co. v. Minnesota, supra, 358 U.S. at p. 460 [ 79 S.Ct. 357 , 3 L.Ed.2d at pp. 428-429]; International Harvester Co. v. Evatt, 329 U.S. 416, 422-423 [ 67 S.Ct. 444 , 91 L.Ed. 390, 395-396 ]; Butler Bros. v. McColgan, supra, 315 U.S. at pp. 506-507 [ 62 S.Ct. 701 , 86 L.Ed. at pp. 995-996]; El Dorado Oil Works v. McColgan, supra, 34 Cal.2d at pp. 744-745; John Deere Plow Co. v. Franchise Tax Board, supra, 38 Cal.2d at pp. 224-225; Hartman, op. cit. supra, 13 Vand.L.Rev. at p. 67.) Precision is not required, but only “rough approximation.” (International Harvester Co. v. Evatt, supra, 329 U.S. at p. 422 [ 67 S.Ct. 444 , 91 L.Ed. at p. 395 ]; El Dorado Oil Works v. McColgan, supra, 34 Cal.2d at p. 741 .)

The techniques of income apportionment, in all their unavoidable complexity, represent a search for meaningful measurements by which to gauge the portion of integrated income-producing activity attributable to a particular taxing jurisdiction. If the business activity is relatively immobile, location or space is the principal dimension to be measured. Some businesses, particularly the transportation industry, have strong migratory as well as stationary aspects. Thus the element of time becomes a logically acceptable dimension, available for measurement. A third dimension, which we might call the “production element,” is present. Activity occurring at a particular location and within a particular time span may have been produced by activity at a different place and earlier time; and it might produce no profit at all except for the fact that it will result in further activity at a third location during a third time span. The production element thus transcends time-space. Economic activity in the transportation industry moves in all three of these dimensions.

Allocation formulae devised for specialized use in the taxation of interstate transportation commonly take account of one or more of these dimensions. (See Braniff Airways v. Nebraska State Board etc. & Assessment, 347 U.S. 590 [ 74 S.Ct. 757 , 98 L.Ed. 967 ] ; Ott v. Mississippi Valley Barge Line Co., 336 U.S. 169 [ 69 S.Ct. 432 , 93 L.Ed. 585 ] ; Central Greyhound Lines v. Mealey, 334 U.S. 653 [ 68 S.Ct. 1260 , 92 L.Ed. 1633 ]; Norfolk & Western Ry. Co. v. North Carolina, supra, 297 U.S. 682 [ 56 S.Ct. 625 , 80 L.Ed. 977 ]; Flying Tiger Line, Inc. v. County of Los Angeles, 51 Cal.2d 314 [ 333 P.2d 323 ]; Slick Airways, Inc. v. County of Los Angeles, 140 Cal.App.2d 311 [ 295 P.2d 46 ]; Pacific Fruit Express Co. v. McColgan, supra, 67 Cal.App.2d 93 ; Altman and Keesling, op. cit. supra, pp. 117-118.) But insistence on measurement of one or two of these dimensions to the exclusion of the remainder rests on a distortion. We must not adopt the attitude of the blind man who seized the elephant’s tail and compared the animal to a rope. It is enough that the formula rest on one or a combination of logically relevant factors. Evidence directed at one of several related factors falls short of demonstrating that the formula is arbitrary or produces an unreasonable result; it “must be directed to each of them alike, for only thus can the assumed relation between them be proved to be unreal.” (Norfolk & Western Ry. Co. v. North Carolina, supra, 297 U.S. at p. 688 [ 56 S.Ct. 625 , 80 L.Ed. at p. 982 ] ; Butler Bros. v. McColgan, supra, 17 Cal.2d at p. 677.)

The notion that an interstate vessel earns income on the high seas, outside any taxing jurisdiction and hence immune from taxation, accepts the time-space dimensions but ignores the “production” dimension. Were it not for the productive potential acquired at the port of departure and liquidated at the port of destination, the high seas portion of the time-space continuum would be barren of profit. The obverse is just as true. The embarkation of payload within one taxing jurisdiction and its discharge within another would be impossible without the connecting voyage over the tax-free face of the deep. The port-day formula acknowledges the productive values supplied to the entire voyage, including the high seas sector, by activities within California.

Both these formulae, voyage-day and port-day, employ the time factor as a qualification upon the spatial elements implicit in the property-payroll-revenue formula. They part company over the production element. The voyage-day method excludes the production element as an extension of shore-based activity accompanying the vessel on the high seas. In insisting upon recognition of California-produced values which contribute to the total value of the entire voyage, the port-day formula is consistent with due process concepts. The very notion of a unitary business implies that the taxpayer’s affairs within and outside the taxing jurisdiction are so indivisible that income from property and activities within the jurisdiction cannot be measured without taking into account income from property and activities located elsewhere. (Butler Bros. v. McColgan, supra, 17 Cal.2d at p. 668; John Deere Plow Co. v. Franchise Tax Board, supra, 38 Cal.2d at p. 223 ; Keesling and Warren, op. cit. supra, 12 Hastings L.J. at p. 46.) The business is not divisible by state lines. It must be treated as a unit even though part of it is in another state or travels on the high seas. In measuring what portion of the sum total is fairly attributable to the taxing state, the latter must necessarily take into account income which, in a sense, has been “earned” outside its borders. There is nothing unconstitutional about this. The only question is whether the state’s method is proportional to the “protection, opportunities and benefits” which it has conferred. (Wisconsin v. J. C. Penney Co., supra, 311 U.S. at p. 444 [ 61 S.Ct. 246 , 85 L.Ed. at pp. 270-271].) Due process does not prohibit a formula which considers out-of-state activity in evaluating the local activity. (International Harvester Co. v. Evatt, supra, 329 U.S. at pp. 420-421 [ 67 S.Ct. 444 , 91 L.Ed. at pp. 393-395]; Hartman, op. cit., supra, 13 Vand.L.Rev. at p. 114.) An argument that the state confers no protection, no opportunities or benefits on the high seas, ignores an imponderable part of the ship’s cargo, the values created at the port of embarkation and port of debarkation. In molding its apportionment formula to embrace productive values supplied at California ports to the entire voyage, the state is taxing not extraterritorial values, but values “reasonably attributable” to activity within California.

Having concluded that the port-day formula is reasonably proportioned to California-contributed values and hence consistent with due process, we turn to the interstate commerce argument. Corporations engaged exclusively in interstate commerce are subject to a net income tax on a portion of their net income reasonably attributable to the taxing state. (Northwestern States Portland Cement Co. v. Minnesota, supra, 358 U.S. 450 [ 79 S.Ct. 357 , 3 L.Ed.2d 421 ]; West Publishing Co. v. McColgan, 328 U.S. 823 [ 66 S.Ct. 1363 , 90 L.Ed. 1603 ], affirming 27 Cal.2d 705 [ 166 P.2d 861 ]; Memphis Natural Gas Co. v. Beeler, 315 U.S. 649 [ 62 S.Ct. 857 , 86 L.Ed. 1090 ]; Norfolk & Western Ry. Co. v. North Carolina, supra, 297 U.S. 682 [ 56 S.Ct. 625 , 80 L.Ed. 977 ] ; Hartman, op. cit., supra, 13 Vand.L.Rev. at p. 64.) The states, however, may not discriminate against interstate commerce by subjecting it to cumulative tax burdens to which local commerce is not exposed. If state taxation imposes on interstate commerce multiple taxes or the risk of multiple taxation, it is discriminatory and must fall. (Gwin, White & Prince v. Henneford, 305 U.S. 434, 439 [ 59 S.Ct. 325 , 83 L.Ed. 272, 276 ] ; J. D. Adams Mfg. Co. v. Storen, 304 U.S. 307, 311 [ 58 S.Ct. 913 , 82 L.Ed. 1365, 1369 , 117 A.L.R. 429, 433 ].)

Gist of the interstate commerce objection is the risk of multiple state taxation upon the ocean voyage through sanctioning use of high seas time as an element in apportionment formulae which might be used by several states. An apportioned tax, measured by the values created in the taxing state, eliminates in a practical sense the danger of multiple tax burdens. (Northwestern States Portland Cement Co. v. Minnesota, supra, 358 U.S. at pp. 462-463 [ 79 S.Ct. 357 , 3 L.Ed.2d at pp. 429-430]; Standard Oil Co. v. Peck, 342 U.S. 382, 384 [ 72 S.Ct. 309 , 96 L.Ed. 427 , 430, 26 A.L.R.2d 1371 , 1374]; see Schwartz, The Supreme Court, pp. 215-217.) True, there is a theoretical possibility that divergent formulae or varying definitions in administration of seemingly identical formulae may create cumulative tax burdens. The taxpayer, however, has not demonstrated any tangible possibility that more than 100 per cent of its unitary income will be taxed. “There is nothing to show that multiple taxation is present. We cannot deal in abstractions. In this type of case the taxpayers must show that the formula places a burden upon interstate commerce in a constitutional sense. This they have failed to do.” (Northwestern States Portland Ce ment Co. v. Minnesota, supra, 358 U.S. at p. 463 [ 79 S.Ct. 357 , 3 L.Ed.2d at p. 430 ].)

Lastly, we reach the objection that by allocating to California a portion of vessel property-payroll-revenue occupied by the high seas voyage, the port-day formula violated statutory language limiting the tax to income “derived from sources within” and resulting from “activities carried on in” this state. Such an interpretation limits ambit of the statute more narrowly than constitutional lines, at least as we view the latter. The statute, on the contrary, is coextensive with the state’s constitutional power to tax. (Butler Bros. v. McColgan, supra, 17 Cal.2d at p. 677 ; see also Matson Navigation Co. v. State Board of Equalization, 3 Cal.2d 1, 8 [ 43 P.2d 805 ].)

In 1939 section 13 of the Corporation Income Tax Act was amended by addition of the following sentence: “Income from business carried on partly within and partly without this State shall be allocated in such manner as is fairly calculated to apportion such income among the States or countries in which such business is conducted.” It has been said that the amendment was designed to sanction the port-day formula by preventing steamship income from being attributed to the high seas. (Altman and Keesling, op. cit. supra, pp. 119-120.) We have no usable interpretive aids pointing to that statutory purpose. The precise language is subject to the somewhat metaphysical objection that business conducted on the high seas cannot be divided up between the states or countries in which it is conducted because it is not conducted within states or countries. Aside from conflicting possibilities suggested by fine-spun exegesis, the amendment’s essential objective is fairly apparent, that is, to allow division of all and not part of income for tax purposes among the various jurisdictions which had power to tax it. It was not designed to deprive the taxing agency of the very flexibility in apportionment delegated to it by the preceding part of the same section. (See El Dorado Oil Works v. McColgan, supra, 34 Cal.2d at p. 736 .) The port-day formula is consistent with the amendment’s objective.

Judgment affirmed.

Sehottky, J., and Janes, J. pro tern., concurred.

Appellant’s petition for a hearing by the Supreme Court was denied October 23, 1963. Traynor J., did not participate therein.

During the tax years in question, 1942 through 1947, the taxing statute was the Corporation Income Tax Act (Cal. Stats. 1937, ch. 765, as amended by Stats. 1939, ch. 1049), which has since been codified as sections 23501 et seq., Revenue and Taxation Code. The function of the California corporation income tax, as distinguished from the corporation franchise tax (Rev. & Tax Code, 5 23101 et seq.), was described in West Publishing Co. v. McColgan, 27 Cal.2d 705 [ 166 P.2d 861 ], Cases sustaining state income taxation based upon an allocated portion of net income derived solely from interstate corporate business are summarized and collected in Northwestern States Portland Cement Co. v. Minnesota, 358 U.S. 450 [ 79 S.Ct. 357 , 3 L.Ed. 2d 421 ]. The Franchise Tax Board points out that it has applied the port-day formula in a restricted manner and only as the vessels, their payrolls and revenue produced unitary income. For example, the port-day method was not applied to shoreside properties, shoreside payroll or bareboat charter revenue. Flexible administration of apportionment formulae to meet individual situations is discussed and sanctioned in El Dorado Oil Works v. McColgan, supra, 34 Cal.2d at pp. 735-736. See Rev. & Tax. Code, § 25101. The port-day formula is discussed in Altman and Keesling, op. cit. supra, p. 120, and in Wahrhaftig, Allocation Factors in Vse in California, 12 Hastings L.J. 65 , 81-82. In 1957 Revenue and Taxation Code section 25101 was amended, and it is said that this amendment eliminated the port-day formula and substituted the voyage-day formula as a statutory mandate. (See Cal. Admin. Code, tit. 18, reg. 25101; Wahrhaftig, op. cit. supra, p. 82.) At the same time section 25101.1 was added, declaring that the Legislature was legislating for tbg future ÍUtgu4e4 U9 euee US to preceding tax years, We do not mean to devalue the factor of location. It may be important or unimportant, according to the characteristics of the particular business enterprise. In Butler Bros. v. McColgan, supra, and in Bass, Ratcliff & Gretton, Ltd. v. State Tax Com., supra, the state was able to tax an allocated portion of the company’s unitary profits even though the company had lost money on its in-state operations. In John Deere Plow Co. v. Franchise Tax Board, supra, 38 Cal.2d at pp. 226-227, the court states: “With regard to a unitary business, the place where expenses affecting the entire business are incurred has little or no bearing on the question of how such expenses should be allocated.” On the other hand, in El Dorado Oil Works v. McColgan, supra, the court sustained a formula which assigned 100 per cent of sales, including interstate sales, to California, since the taxpayer had no sales facilities outside this state, saying ( 34 Cal.2d at p. 742 ) : “Accordingly, the focal point for consideration is the place where the activities of the corporation occurred which resulted in the sales. ’ ’ In Wisconsin v, J. C. Penney Co., supra, the very event upon which the apportioned tax was levied, a declaration of corporate dividends, took place outside the taxing state, and this fact generated the following statement ( 311 U.S. at p. 445 [ 61 S.Ct. 246 , 85 L.Ed. at p. 271 ]) : “The fact that a tax is contingent upon events brought to pass without a state does not destroy the nexus between such a tax and transactions within a state for which the tax is an exaction.” Thus, in the formula taxation of unitary interstate income, physical location signals neither tax immunity nor subjection. During the tax years in question the applicable sections of the Corporation Income Tax Act (Cal. Stats. 1937, ch. 765, as amended by Stats. 1939, ch. 1049) read as follows: “§ 3. There shall be levied, collected and paid for each taxable year, a tax . . . upon the net income of every corporation derived from sources within this State. . . . Income from sources within this State includes income from tangible or intangible property located or having a situs in this State and income from any activities carried on in this State, regardless of whether carried on in intrastate, interstate or foreign commerce. ’ ’ “ f> 13. In the case of corporations owning property or engaging m business or activities both within and without the State, the net income derived from sources within this State shall be determined by an allocation upon the basis of sales, purchases, expenses of manufacture, pay roll, value and situs of tangible property or by reference to any of these or other factors or by such other method of allocation as is fairly calculated to determine the net income from sources within this State. Income from business carried on partly within and partly without this State shall be allocated in such manner as is fairly calculated to apportion such income among the States or countries in which such business is conducted. ’ ’ Assigned by Chairman of Judicial Council.