32 CHARGES OF GREATEST EFFICACY FOR ARTILLERY AT SEA. 



the impinging spheres; because the surfaces of spheres 



are as the squares of their diameters ; that is — 7- nz — X 



fn r 



5! 



^ . , M F ,4,, . 



But m general — =: — r X — — : Therefore 

 m / ^r 



equating these two values of the whole resisting forces We 



F Q R D* 'f R D* 



have _ X - =: - X ^ and - = ~ X -^^ 



X '4r ; and since the quantities of matter in spheres are 



Q 

 in the conjoint ratio of their magnitudes and densities, 

 or of the cubes of their diameters and densities; it is 



^ - ?i 5! ii ^ - 5: ^ 



7 "" 7" ^ (i* ^ D» ^ N ~ r ^ D ^ 



ft 



— : that is the forces retarding spheres penetrating uni- 



forrhly resisting substances are as the absolute strengths of 

 the fibres of the substances directly, arid the diameter and 

 specific gravities of the spheres inversely* 



Q. E. D. 



Lemma ll. 



Law of the The whole spaces or Depths to which Spheres impinging 



«lepth to which ^^ diffietentty resisting substances penetrate, are as the 



netrate. Squares of the initial Velocities, the Diameters and sped- 



fc Gravities of the Spheres directly, and the absolute 



S 



- Strengths of the resisting Substances inversely : or — 



_ V* D ^ r 



- V ^ 'd ^ n ^ n' 



Proof. For by mechanics — = -»■ X 4r ; and by the pre* 



s V " 



ceding Lfemma ~ = r^ X -7 X — ; thereforebysub- 

 ^ K a n 



S V* D N r 



atitution —zr-^X-rX — X-tS. 



s w* d n R 



Q. E. t^. 



These' 



