  This paper contains a non-trivial generalization of the Harish-Chandra
transforms on a connected semisimple Lie group $G,$ with finite center, into
what we term spherical convolutions. Among other results we show that its
integral over the collection of bounded spherical functions at the identity
element $e \in G$ is a weighted Fourier transforms of the Abel transform at
$0.$ Being a function on $G,$ the restriction of this integral of its spherical
Fourier transforms to the positive-definite spherical functions is then shown
to be (the non-zero constant multiple of) a positive-definite distribution on
$G,$ which is tempered and invariant on $G=SL(2,\mathbb{R}).$ These results
suggest the consideration of a calculus on the Schwartz algebras of spherical
functions. The Plancherel measure of the spherical convolutions is also
explicitly computed.
