SEC. XII.] OF NATURAL PHILOSOPHY. 229 PS3 + SE2 + 2PSD. But because the triangles SPH, SHI are alike, SE2 or SH2 is equal to the rectan gle PSI, Therefore PE2 is equal to the rectangle contained under PS and PS -f SI + 2SD ; that is, under PS and 2LS + 2SD ; that is, under PS and 2LD. Moreover DE2 is equal to SE2 — SD% or SE2 — LS2 + 2SLD — LD2, that is, 2SLD — LD2 — ALB. For LS°- — SE2 or LSa — SAa (by Prop. VI, Book II, Elem.) is equal to the rectan gle ALB. Therefore if instead of DE 2 we write 2SLD — LD 2 — ALB, the quantity -™ - ^-, which (by Cor. 4 of the foregoing Prop.) is as PE x the length of the ordinate DN, will 2SLD x PS LD2 X PS now resolve itself into three parts ALB xPS ... -TE3rr~ -pfixT" -pE^-v-;whereifinsteadofVwewnt the inverse ratio of the centripetal force, and instead of PE the mean pro portional between PS and 2LD, those three parts will become ordinates to so many curve lines, whose areas are discovered by the common methods. Q.E.D. EXAMPLE 1. If the centripetal force tending to the several particles of the sphere be reciprocally as the distance ; instead of V write PE the dis tance, then 2PS X LD for PE2 ; and DN will become as SL — £ LD — •ny