VOL. XXII.] PHILOSOPHICAL TRANSACTIONS. 543 



stration depends on a general theorem, which is easily deduced from Prop. 35. 

 Book 2. Newton's Principles, &c. 



Carol. 1. — Hence, if bg, bg, be infinitely small parts of the lines ag, Ag, 

 and bB be produced to l ; then, the resistance against bg (which call e) will be 

 to the resistance against bg (which call e) as gl^ to gb*. 



For, e : E :: Knbg : Pfibg, that is, e : e :: bg X bn : bg X bs (by the pre- 



C M ^ 



ceding lemma) ; therefore, e : e :: bn : bE; that is, e : e :: — : bc (by the 



construction in the above lemma) : therefore, e : e :: cm* : bc". But, by reason 

 of the similar triangles bmc, glb, cm^ : bc^ :: gl* : gb^ ; therefore e : e ;: 

 GL^ : gb\ q.e.d. 



Carol. 1. — The resistance against an infinitely small part gb, is expressed by 

 the cube of the line gl divided by the square of the line gb. For, if all the 

 infinitely small parts of the line Ag, as bg, be supposed equal, then the resist- 

 ance against bg, may be expressed by the same bg, that is, e := bg, which is 

 the same as e = gl. Therefore, by Carol. 1, e : gl :: gl* : gb% whence e 



= . Q.E.D. 



Coral. 3. — If r be the radius and c the circumference of any circle, then the 

 resistance against the conic surface, generated by the rotation of the small line 



gb about Ai, is equal to the product of into — ^. For the resistance 



against the conic surface is equal to all the resistances of the lineolae gb, or to 

 all the es, that is, equal to the circumference of the circle whose radius is 

 BM, multiphed into e ; that is, the resistance against such a conic surface is 



GL' 



X e ; which, by Cor. 2, is = X — ; 



GB- 



Problem I. — To discover the curve line whose rotation shall produce a solid, 

 which while it is moving in a fluid medium according to the direction of its 

 axis, shall meet with the least resistance. 



Let OG, GB, (fig. 2, pi. 13,) be two infinitely small particles in the required 

 curve, which by revolving about aq will produce the solid of least resistance. 

 Draw BM, GP, perpendicular to aq, likewise bl and gn to aq, and on parallel 



»T- C X BM X gl' ... . 1 r 1 



to BM. JNow — ^. ^^i — is the resistance against the surface generated 



by the rotation of gb about aq, and ^ ^ "^ is the resistance against the 



' ' r X OG- ° 



surface generated in like manner by og. {Cor. 3.) Now the resistance to 

 both these together should be a minimum, that is to say, *" ^^'x'gb''^'^ + 



C X GP X 0N3 . . „, . . , ^ I • I 1- 



— J. ^ ^jgj — = a minimum. ihis is as much as to find, in the hne ks so 



