# Page:CarmichealMass.djvu/8

Remembering that ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ are the longitudinal mass and transverse mass, respectively, and making use of theorem I., we have the following result:

Theorem II. Let ${\displaystyle m_{0}}$ denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S denote by ${\displaystyle l\left(m_{v}\right)}$ its longitudinal mass, that is, its mass in a direction parallel to its line of motion. Then we have

${\displaystyle l\left(m_{v}\right)={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}}$

where ${\displaystyle \beta =v/c}$ and c is the velocity of light ${\displaystyle \scriptstyle \left(MVLRC_{1}C_{2}\right)}$.

## § 3. On the Dimensions of Units.

Denote the fundamental measurable physical entities mass, length and time by M, L and T respectively. Then the definition of derived entities gives rise to the so-called dimensional equations. Thus if B denote velocity, then from the definition of velocity we have the dimensional equation

${\displaystyle V={\frac {L}{T}}}$.

That such equations must be useful in obtaining the relations of a derived unit in two systems of reference is obvious. Thus from the above dimensional equation for V we may at once derive the fundamental result (see previous paper, theorem VI., p. 170) concerning the relation of units of length in the line of relative motion of two systems not at rest relatively to each other. For this purpose it is sufficient to employ postulate V. and theorem IV. of the previous paper. The reader can easily supply the argument. Or, conversely, if one knows the relations which exist between units of length and units of time in two systems one concludes readily to the truth of postulate V.

Likewise, from the dimensional equation

acceleration = ${\displaystyle {\frac {L}{T^{2}}}}$

one may readily determine the relations which exist between units of acceleration on two systems, it being assumed that the relations of time units and length units are known. Making this assumption, then, the two dimensional equations above give us the following theorem:

Theorem III. Let two systems ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ move with a relative velocity v in the direction of a line l, and let ${\displaystyle \beta =v/c}$ where c is the velocity of light. Then to an observer on ${\displaystyle S_{1}}$ it appears that the unit of velocity [acceleration] on