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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.
{tex}\therefore{/tex} M₁ = 1 kg, M₂ = {tex}\alpha{/tex} kg
L₁ = 1 m, L₂ = {tex}\beta{/tex} m
T₁ = 1 s, T₂ = {tex}\gamma{/tex} 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 {tex}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] }{/tex}
{tex}= 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 }{/tex}
{tex}= 4.2 \left[ \frac { 1 } { \alpha } \right] \times \left[ \frac { 1 } { \beta } \right] ^ { 2 } \times \left[ \frac { 1 } { \gamma } \right] ^ { - 2 }{/tex}
n₂ = 4.2{tex}\alpha ^ { - 1 } \beta ^ { - 2 } \gamma ^ { 2 }{/tex} new unit
Therefore, 1 cal = {tex}4.2 \alpha ^ { - 1 } \beta ^ { - 2 } \gamma ^ { 2 }{/tex} new unit

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