Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group  G=Gal(F_p²K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern  AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants  0<d<10⁸ and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F₃²K∣K) and the 3-class field tower groups G=Gal(F₃^∞K∣K).