230 THE MATHEMATICAL PRINCIPLES [BOOK 1 the hyperbolic crrve ab. And the chord ba being drawn, will inclose the area aba equal to the area sought ANB. EXAMPLE 2. If the centripetal force tending to the several particles of the sphere be reciprocally as the cube of the distance, or (which is the same PE3 thing; as that cube applied to any given plane ; write 2PS X LD for PE2 ; and DN will become as '2AS2 SL X AS2 for V, and AS2 ALB X AS2 2PS X LD2 LSI PS X LD 2PS that is (because PS, AS, SI are continually proportional), as ALB X SI 2LD: LSI If we draw then these three parts into th length AB, the first r-pr will generate the area of an hyperbola ; the secL-t \J , ALB X SI . ALB X SI ond iSI the area } AB X SI ; the third 2Ll^ — area - 2LA , that is, !AB X SI. From the first subduct the sum of the 2LB second and third, and there will remain ANB, the area sought. Whence arises this construction of the problem. At the points L, A, S, B, erect the perpendiculars L/ Aa Ss, Bb, of which suppose Ss equal to SI ; and through the point s, to the asymptotes L/, LB, describe the hyperbola asb meeting the perpendiculars Aa, Bb, in a and b ; and the rectangle 2ASI, subducted from the hyberbolic area AasbB, will B leave ANB the area sought. .. ,, . „ EXAMPLE 3. If the centripetal force tending to the several particles of the spheres decrease in a quadruplicate ratio of the distance from the parpT^4 _ tides ; write ~