Can I use a 650cc tire on a 27x1 1/4 rim?
No. A 27 inch tire has a 630 mm bead seat diameter, 650c has a 571 mm bead seat diameter. That means the wire/kevlar bead of the tires has a diameter of 571 mm, and the inside of the rim has a diameter of 630 mm. The bead of the tire is designed to have very little stretch (so it stays on the rim at high pressure), so there is no way that you could fit 650c tires on a 27 inch wheel.

(also, see Can I put 700c tires on rims made for 27 x 1.25&quot;? for a case where you might be able to squeeze smaller tires on a larger rim—but definitely shouldn't.)