§72. Hierarchy of Superscripts and Subscripts: Superscripts or subscripts may carry superscripts or subscripts of their own; the latter are then referred to as superscripts or subscripts of second order, and are thus distinguished from the former, which are called superscripts or subscripts of first order. Second order superscripts or subscripts may, in turn, carry superscripts or subscripts of their own, which are then called superscripts or subscripts of third order. While it is theoretically possible for a superscript or subscript to be of order higher than the third, this situation rarely arises in practice.

§73. Level Indicators: A level indicator other than the base-line indicator identifies the symbols which follow it as representing a superscript or subscript. The base-line indicator identifies the symbols which follow it as representing signs on the base line. The degree of elevation or depression specified by a level indicator is always with respect to the base line; the symbol which precedes the indicator, if it represents a sign at some other level, plays no role in this regard.

§74. Orientation by Level Indicator:

a. The effect of a level indicator with one component is to direct the reader's attention upward or downward from the base line according as that component is the superscript or the subscript indicator.

b. The effect of a level indicator with two components may be analyzed as follows:

c. The effect of a level indicator with three components may be analyzed as follows:

d. The effect of a level indicator with more than three components may be analyzed in the same manner suggested for level indicators with two or three components.

§75. Left Superscripts and Subscripts: A superscript or subscript may occupy a position to the left, as well as to the right, of the sign to which it applies. The words left or right are then used with the words superscript or subscript to make the distinction in position.

            A right or left superscript or subscript is represented as such merely by preserving the relative horizontal positions of the superscript or subscript symbol to which it applies. Each must be preceded by its appropriate level indicator.

            Left superscripts or subscripts of the third or higher order, although rare, are treated in the manner suggested by the examples below.

§76. Direct Superscripts and Subscripts: A superscript or subscript which occupies, respectively, a position directly over or under the sign to which it applies is called a modifier (see Rule XIV).

§77. Numeric Subscripts: The subscript indicator must not be used to indicate a numeric subscript provided that all of the following conditions hold:

§78. Comma at Superscript or Subscript Level: A commonly occurring superscript or subscript notation is the one in which two consecutive items are separated by a comma with an optional space following the comma. In this configuration, the symbol ?? (dots 2-4-6) must be used to replace the comma and the optional space used in this way. This contracted form must not be used to replace a comma and the optional space which follows it in a configuration which is on the base line.

§79. Circumstances Determining Changes of Level: The symbols and situations listed below have the following effect in determining changes of level.

a. A level indicator terminates the effect of a previous level indicator and initiates the level implied by the new indicator. In the case of the base-line level, the previous base-line indicator may only be implied.

b. The punctuation indicator terminates the effect of any previous level indicator and initiates the base-line level. In addition, the comma, provided it is not a numeric symbol, terminates the effect of any previous indicator and initiates the base-line level. However, the comma, when it is a numeric symbol and the contracted form ?? (dots 2-4-6), preserves the level that is already in effect.

c. A space or the transition to a new braille line which is followed by literary text or unrelated mathematical text terminates the effect of any previous level indicator and initiates the base-line level. However, if a space occurs between the parts of an abbreviation or phrase, the appropriate level indicator must be restated before each part.

d. The space which immediately follows a symbol of shape, an abbreviated function name, or an unabbreviated function name, provided the latter is in a mathematical context, preserves the level that is already in effect. If these items carry a superscript or subscript, the space which follows such a superscript or subscript reinstates the level that was in effect before.

e. The space which occurs in a numeral for the purpose of dividing it into short regular segments preserves the level already in effect.

f. The space which precedes an ellipsis or long dash preserves the effect of any previous level indicator. The space which follows the ellipsis or long dash preserves the level that is already in effect. However, if such a space is followed by literary text, unrelated mathematical text, or a sign of comparison, this space initiates the base-line level.

g. The space or transition to a new braille line which is followed by a comparison symbol terminates the effect of a level indicator already in effect and initiates the base-line level. The space after a comparison symbol preserves the level that is already in effect.

h. Any symbol or situation other than those in a to g preserves the level that is already in effect.

§80. Use of Level Indicators:

a. A level indicator must be used before any braille indicator or grouping symbol whenever this braille indicator or grouping symbol applies to a level other than the one currently in effect.

b. The superscript indicator must be used to restate the superscript level when two superscripts are consecutive but one applies to the expression which precedes it and the other applies to the expression which follows it. Similarly, the subscript indicator must be restated when two subscripts are consecutive and one applies to the expression preceding it and the other applies to the expression following it. A superscript or subscript indicator must be restated before a modified expression which is interior to the superscript or subscript expression, provided that the multipurpose indicator is also used.

c. The appropriate level indicator must be used before each part of an abbreviation or phrase which is at a level other than the base line.

d. Whenever spaces are left for the purpose of achieving alignment, level indicators must be used as though such spaces were not present.

e. The appropriate level indicator must be used before any symbol or situation in which a change of level is required but the change is not effected by any of the conditions of §79.

§81. Non-Use of Level Indicators:

a. The base-line indicator must not be used to return to the base line from a numeric subscript if the subscript indicator has not been used before the numeric subscript.

b. The base-line indicator must not be used before a right enlarged grouping symbol if this symbol either is separated from its preceding material by one or more spaces, or if the material which precedes the right grouping symbol is not the end of an expression.

c. A level indicator must not be used before any closing grouping symbol which is drawn in.

d. A level indicator must not be used to change the level if any symbol or situation specified in §79 has already effected the change to the desired level.

§82. Simultaneous and Non-Simultaneous Superscripts and Subscripts:

a. When an expression simultaneously carries a superscript and subscript, the subscript must be indicated first, even if the subscript is numeric and does not require the subscript indicator. However, if this sign carries one or more primes in addition, see §83.

b. When the same expression carries a superscript and a subscript which are not simultaneous, the relative horizontal position of the signs must be retained in the transcription, but the base-line indicator must be inserted before making the transition to the other level.

§83. Primes in Addition to Superscripts or Subscripts:

a. The prime symbol must never be preceded by the superscript indicator.

b. When an expression carries one or more primes in addition to superscripts or subscripts, the prime symbol or symbols must be indicated first unless such symbols do not occur at the beginning of the superscript or subscript, in which case they must retain the same position as in ink print.

c. For primes in other roles see §172.

§84. Plurals and Possessives: For plurals or possessives of mathematical expressions which end with a superscript or subscript see §39.