Two-dimensional circulatory flow about an internal boundary immersed in infinite fluid 

 leads to an infinite value for T^. This circumstance, altliough inconvenient, does not invali- 

 date other conclusions from the theory, since a cylinder of infinite length is in any case an 

 abstraction introduced in order to simplify the mathematics. 



18. UNITS OF MEASUREMENT 



In all cases, a consistent set of dynamical units is assumed to be employed. In using 

 each formula, any unit of length may be used, but the same unit must be used for all linear 

 dimensions, the square of that unit must be used for areas, and the cube for volumes. A 

 common unit of time must be employed for all velocities and accelerations. 



If forces are measured in pounds, time in seconds, and linear dimensions in feet, then 

 pressure is in pounds per square foot; mass is measured in slugs, or pounds times seconds 

 squared divided by feet, and equals weight in pounds divided by the acceleration of gravity or 

 by 32.2; density is in slugs per cubic foot; energy is in foot-pounds. 



If forces are measured in pounds and time in seconds, but linear dimensions in inches, 

 then pressure is in pounds per square inch; mass is in pounds times seconds squared divided 

 by inches and equals weight in pounds divided by 386, which is the acceleration due to 

 gravity expressed in inches per second squared; density equals pounds per cubic inch divided 

 by 386; energy is in inch-pounds. 



The velocity potential has the dimensions of velocity multiplied by distance; hence it 

 will be in feet squared divided by seconds if lengths are expressed in feet and time in seconds, 

 or in inches squared divided by seconds if inches are substituted for feet. The same units 

 apply to the circulation as to the velocity potential. 



Angles may always be measured in radians, and this unit is always understood when an 

 angle is added or equated to a quantity that is not an angle, as in Equation [138f '] in Section 

 138: this holds whether the angle is represented by a single synjijaol, such as d, or indirectly 

 by a symbol such as sin""^. In equations between angles, like Equation [38b], or when a 

 trigonometric function such as sin d is indicated, degrees may be used instead of radians. 



It may be remarked that the symbol \/P^will be used to denote the positive square root 

 of any expression P whenever P represents a positive real number; and such angles as sin~ x 

 or tan~^ar will be understood to be in the first quadrant whenever x is so limited by the circum- 

 stances of the case that this interpretation cannot fail. Otherwise these symbols are to be 

 interpreted as many-valued except insofar as a special rule is stated for their interpretation. 



32 



