s these, moreover, are the very people who will think
themselves privileged to criticise and use their privilege with the
least discretion, I cannot recommend too much clearness, fulness,
and order in the _expose_ of the principles. Were I you, I would
devote to this first part at least double the space you have done.
Your familiarity with the results and formulae has led you into what
is extremely natural in such a case--a somewhat hasty passing over
what, to a beginner, would prove insuperable difficulties; and if I
may so express it, a sketchiness of outline (as a painter you will
understand my meaning, and what is of more consequence, see how it
is to be remedied).

You have adopted, I see, the principle of virtual velocity, and the
principle of d'Alembert, rather as separate and independent
principles to be used as instruments of investigation than as
convenient theories, flowing themselves from the general law of
force and equilibrium, to be first _proved_ and then remembered as
compact statements in a form fit for use. The demonstration of the
principle of virtual velocities is so easy and direct in Laplace
that I cannot imagine anything capable of rendering it plainer than
he has done. But a good deal more explanation of what _is_ virtual
velocity, &c., would be advantageous--and virtual velocities should
be kept quite distinct from the arbitrary variations represented by
the sign [Greek: d].

With regard to the _principle of d'Alembert_--take my advice and
explode it altogether. It is the most awkward and involved statement
of a plain dynamical equation that ever puzzled student. I speak
feelingly and with a sense of irritation at the whirls and vortices
it used to cause in my poor head when first I entered on this
subject in my days of studentship. I know not a single case where
its application does not create obscurity--nay _doubt_. Nor can a
case ever occur where any such principle is called for. The general
law that the change of motion is proportional to the moving force
and takes place in its direction, provided we take care always to
regard the _reaction_ of curves, surfaces, obstacles, &c., as so
many real moving forces of (for a time) unknown magnitude, will
always help us out of any dynamical scrape we may get into. Laplace,
page 20, Mec. Cel. art. 7, is a li