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We consider a vector bundle $E\to M$ and the principal bundle $PE$ of frames of $E$. We determine all natural transformations of the connection bundle of the first order principal prolongation of principal bundle $PE$ into itself.

We consider a vector bundle $E\to M$ and the principal bundle $PE$ of frames of $E$. Let $K$ be a principal connection on $PE$ and let $\Lambda$ be a linear connection on $M$. We classify all principal connections on $W^{2}PE=P^{2}M{\times }_M{J}^{2}PE$ naturally given by $K$ and $\Lambda$.