RESISTANCE. 



body ; for it is evident by what we have faid, that this can 

 only be confidered as nearly true in fmall variations of ve- 

 locity, and can never be applied in the comparing together 

 of the reiiftances to all velocities whatever, without the molt 

 enormous errors. See New Principles of Gunnery, by 

 Mr. Robins, chap. 2. prop. i. See Resistance of the A'tr- 

 See alfo Projectile and Gunnery. 



Refiftance and retardation are ufed indifferently for each 

 other, as being both in the fame proportion, and the fame 

 refinance always generating the fame retardation. But, 

 with regard to different bodies, the fame refiftance frequently 

 generates different retardations ; the refiftance being as the 

 quantity of motion, and the retardation as that of the ce- 

 lerity. For the difference and meafure of the two, fee Re- 

 tardation. 



The retardations from this rcfiltance may be compared 

 together, by comparing the refiftance with the gravity or 

 quantity ot matter. Thus, let v = the velocity ; a = the 

 area of the face, or end of a cylinder ; n = the fpecific 

 gravity of the fluid ; g = 32,; feet, the force of gravity ; 



then the altitude due to the velocity i> being — , the whole 



refiftance, or motive force m, will be exprefled by the fol- 

 lowing formula: ; 



•u z anv'' anv z s' < 



m = an x — ■ = ; or = — : 



2 S 2 g i S 



the latter having place when the motion is not in a direc- 

 tion perpendicular to the plane or end, but is inclined to it 

 at a given angle, whofe fine is j-. For it is a known pro- 

 perty, that, in this cafe, the refiftance varies as the cube of 

 the fine of the angle of inclination. If now 10 be made to 

 denote the weight of the body, and / the retarding force ; 

 then, on the fame principles, we derive 



m anv'' s 1 



iv zg nv 



If the body be a cylinder, moving in the direftion of its 

 axis, and the diameter of the bafe equal d, or radius r, and 

 ~ = 3.14159, &c. then 



m irnd''v z <x n r' 11' 



iu Sg iu 2 g 111 



But when the cylinder moves in a direftion perpendicular to 

 its axis, then writing h for the height or length of the cy- 

 linder, we have 



"hd nv'hr 



m nv 



■w ' sgnv 



vjfw 



And when it moves obliquely to its axis, then writing* for 

 the fine of the angle of inclination, we have 



rbs* r u'- 1 . q ( -r 2 — i) 1 



/ = 



2 S 



{ ' - *T= 



+ 



40 



+ 



+ 



1 JBt : 



&c. I + ■ 



j 



2 S 



.(-x 1 ) 1 



See Moore's Theory of Military Rockets. 



If the body be a cone, then the fame notation remaining, 

 only writing s for the fine of the angle of inclination of 

 the fide of the cone : then 



/=- = 

 iv 



vnd> 



itnr v's 



8 g tv 2 g iv 



For, in this cafe, the inclination has no effeft in re- 

 ducing the fe&ion oppofed to the refiftance of the fluid, 

 this being the fame as in the cylinder, and therefore will 

 vary a.s s '•'. 

 Vol. XXX. 



The fame notation ftill remaining, it ]s found, from a flux - 

 ional inveftigation (fee Gregory's Mechanics, vol. i.) 

 the refiftance of the body, when terminated with a hemi- 

 fpherical furface, is 



_ m irnd'v* ibc'j 1 



tv 16 g iv \S"U> 



that is, half what it is when the end is a plane furface. 



Hence the refiftance of a fphere, when impelled through 

 any fluid, is equal to half the direct refiftance to a great 

 circle of it, or to a cylinder of the fame diameter. Since 

 £ 7T d' is the magnitude of the globe, if N denotes its den- 

 fity, or fpecific gravity, its weight m = ^uNi/ 1 ; and. 

 therefore, the retardive force becomes 





rr n i< z d ' 

 l6g iv 



tN</ ! 



3««'* 



SgNd~ zgs' 



where s is the fpace defcribed ; for zfgs = v\ by the law 

 of accelerated or retarded motions. From which it appears, 

 that the refiftance varies as the fquare of the velocity di 

 redly, and as the diameter inverfely, all things elfe being 

 the fame ; and hence the reafon, why a large ball overcomes 

 refiftance better than a fmaller one. 



James Bernoulli demonftrates the following theorems, 

 Afta Erud. Lipf. for June 1693, P* 2 5 2 ' & c * 



1. If an ifofceles triangle be moved in a fluid according 

 to the direftion of a line perpendicular to its bafe ; firft, 

 with the vertex foremoit, and then with its bafe; the refin- 

 ances will be in the duplicate ratio of the bafe, and of the 

 fum of the legs. 



2. The refiftance of a fquare, moved according to the di- 

 reftion of its fide, is to the refiftance of the fame fquare, 

 moved with the fame celerity in the direftion of its diagonal, 

 as the diagonal is to the fide. 



3. The refiftance of a circular fegment, lefs than a femi- 

 circle, carried in a direftion perpendicular to its bafis, when 

 it goes with the bafe foremoft, and when with its vertex 

 foremoft (the fame direftion and celerity continuing), is as 

 the fquare of the fame diameter to the fame, lefs one- 

 third of the fquare of the bafe of the fegment. Hence, 

 the refiftances of a femicircle, when its bafe and when its 

 vertex go foremoft, are to one another in a fefquialterate 

 ratio. 



4. A parabola moving in the direftion of its axis, firft 

 with its bafis, and then its vertex foremoft, has its refin- 

 ances as the tangent to an arc of a circle, whofe diameter 

 is equal to the parameter, and the tangent equal to half the 

 baiis of the parabola. 



5. The refinances of an hyperbola and ellipfis, when the 

 vertex and bafe go foremoit, may be thus computed. Say, 

 as the fum ■(or difference! of the trjnfverfe axis and latus 

 reftum is to the tranfverie axis, fo is the fquare of the latu* 

 rectum to the fquare of the diameter ot a certain circle ; 

 in which circle apply a tangent, equal to half the bafis of 

 the hyperbola or ellipfe. Then fay again, as the fum (or 

 difierence) of the axis and parameter is to the parameter, 

 fo is the aforefaid tangent to another right line. And 

 farther, as the fum (or difference) of the axis and parame- 

 ter is to the axis, fo is the circular arc correfponding to the 

 aforefaid tangent to another arc. This done, the refin- 

 ances will be as the tangent to the fum (or difference) of the 

 right line thus found, and the arc lalt mentioned. 



6. In the general, the refinances of any figure wlutever, 

 going now with its bafe foremoit, and then with its vertex, are- 

 as the figures of the bafe to the fum of all the cubes of the 

 elements of the bafe, divided by the fquares of the elements 

 of the curve line. 



E AU 



