  We show that if we factor the long exact sequence in cohomology of a cofiber
sequence of spectra into short exact sequences, then the $d_2$-differential in
the Adams spectral sequence of any one term is related in a precise way to
Yoneda composition with the 2-extension given by the complementary terms in the
long exact sequence. We use this to give a complete analysis of the Adams
spectral sequence for the connective image-of-$J$ spectrum, finishing a
calculation that was begun by D. Davis in 1975.
