TY - JOUR AU - Pars, L. A. AB - INEQUALITIES IN THE RESTRICTED PROBLEM OF THREE BODIES. 355 INEQUALITIES OCCURRING IN THE RESTRICTED PROBLEM OF THREE BODIES L. A. PARS*. We consider a function U of the independent variable x defined by the formula 2P r s where 0, a>j8>0, r = \x+a\, 8 = \x—b\, a= a+ p a-f-p This function occurs in the study of the restricted problem of three bodies, and its properties are important in the classification of the orbits of the planetoid f. The origin 0 is the centre of gravity of a mass a at A (x= —a) b), and I is the length AB. It is clear that U has and a mass fi at B (x = infinities at x = — a and at x = b, and that U->oo with | x Also (except at the anomalous points A and B) 2q 2ft, -r 3 + 3 r 5 dx* 2 2 m eac n so d U/dx > 0, and dU/dx increases steadily from — oo to + °° °f the three regions & (— oo< x < —a), ^ {—a