The calorie is a unit of …

The dimensional formula of energy or heat = u = [ ML²T^-2 ].
Assume that, M₁, L₁, T₁ and M₂, L₂, T₂ are the units of mass, length and time in given two systems.
$\therefore$ M₁ = 1 kg, M₂ = $\alpha$ kg
L₁ = 1 m, L₂ = $\beta$ m
T₁ = 1 s, T₂ = $\gamma$ s
n₁ and n₂ are the magnitudes of heat or energy of system one and system two respectively.
For a given physical quantity, the product of its magnitude and unit is always constant.
n₁u₁ = n₂u₂
or $n _ { 2 } = n _ { 1 } \frac { u _ { 1 } } { u _ { 2 } } = 4.2 \times \frac { \left[ \mathrm { M } _ { 1 } \mathrm { L } _ { 1 } ^ { 2 } \mathrm { T } _ { 1 } ^ { - 2 } \right] } { \left[ \mathrm { M } _ { 2 } \mathrm { L } _ { 2 } ^ { 2 } \mathrm { T } _ { 2 } ^ { - 2 } \right] }$
$= 4.2 \left[ \frac { \mathrm { M } _ { 1 } } { \mathrm { M } _ { 2 } } \right] \times \left[ \frac { \mathrm { L } _ { 1 } } { \mathrm { L } _ { 2 } } \right] ^ { 2 } \times \left[ \frac { \mathrm { T } _ { 1 } } { \mathrm { T } _ { 2 } } \right] ^ { - 2 }$
$= 4.2 \left[ \frac { 1 } { \alpha } \right] \times \left[ \frac { 1 } { \beta } \right] ^ { 2 } \times \left[ \frac { 1 } { \gamma } \right] ^ { - 2 }$
n₂ = 4.2$\alpha ^ { - 1 } \beta ^ { - 2 } \gamma ^ { 2 }$ new unit
Therefore, 1 cal = $4.2 \alpha ^ { - 1 } \beta ^ { - 2 } \gamma ^ { 2 }$ new unit