In this paper, firstly we have derived and presented the specific heat jump as a function of the critical temperature. Secondly, we have analyzed the sign and magnitude of the derived specific heat jump and predicted the expected experimental results. And lastly, we have compared the expected experimental results with the real experimental results. Theoretically derived specific heat jump is considerably compatible with the specific heat jump up and down phenomena of the $YNi_2B_2C$ crystal. Especially, the remarkable theoretical prediction-hat the specific heat would jump down during the normal state-to-superconducting state transition at extremely low temperatures-have been confirmed by the experimental results.