Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/411

Rh the lines themelves PD, QD, generated by thoe increments, will be as the ines of incidence and emergence to each other, and é contra.

The ame thing uppoed, if round the axis AB (Pl. 25. Fig. 10) any attractive uperficies be decribed as CD, regular or irregular, through which the bodies iuing from the given place A mut pas; it is required to find a econd attractive upercies EF, which may make thoe bodies converge to a given place B.

Let a line joining AB cut the firt uperficies in C and the econd in E, the point D being taken any how at pleaure. And upposing the ine of incidence on the firt uperficies to of emergence from the ame, and the ine of emergence from the econd uperficies to the fine of incidence on the ame, to as any given quantity M to another given quantity N; then produce AB to G, o that BG may be to CE as M - N to N; AD to H, o AH may be equal to AG; and DF to K o tha DK may be to DH as N to M. Join KB, and about the centre D with the interval DH decribe a circle meeting KB produced in L, and raw BF parallel to DL; and the point F will couch the line EF, which being turned round the axis AB will decribe the uperficies ought. Q. E. F.

For conceive the lines CP, CQ to be every where perpendicular to AD, DF and the lines ER, ES