*tl 



GUNNERY. 



GUNNEBY. 



(71 



B il'X 



From the firrt of these equation! we have di ; ; which, being 

 substituted in the Utter, give* 



t<lti\ ds 

 rf*y; or, after reduction, ** = -d ^) 



But, multiplying the above equation for iff by 11 A 1 , it become* 

 di (It df= H dt'tpj- ; which, on substituting in the first member 

 the preceding value of ill', become* 







Cdy'\ 

 1 + dJPI' tnerefore * I'M* 

 equation may be put in the form 



This equation, being integrated, would determine the relation between 

 z and y in the trajectory ; and, since dt is constant, the integral of the 



econd member is evidently, . H ^ To obtain that of the first mem- 



- 



ber, let t represent the tangent of half the angle made by a horizontal 

 line and a tangent to the curve at each point ; then, by trigonometry, 



dy 2 t 



we shall have -7- ( = the tangent of the whole angle) = r and 



id\ 2(l + t) ( dy*\ 1 + t* 



d (fo) = (l^tT* d t; "^ *V + rf^J = nt*~ : conaoquentiy, that 



2(l+f)rft t + t> 



first member becomes _ , whose integral is ^_ 



,. (ft t + t' 1 + t 



+ J f~t > or (1 1) + 4 log- J^t ; therefore the integral of equa- 

 tion (III) is 



t + t' 1+t ugdP 



- TS? 



where c is an arbitrary constant. 



Now, let B represent the angle of elevation at the point A, or the 

 angle CAB; at this point we have dx=di cos K, and da? = dz* cos' E; 

 also, at the same point, dz = v dt, v being the initial velocity of the 

 shot. Therefore 



dP 1 



da? = v* dP cos* K, and -^3- = yl ^^ ; but since, b 



V* = 2jh, we have evidently, 



dP 1 



< V >> 



then, if this value be substituted in (IV), and tan 4 E be put for t, 

 that equation will become 



tan J i + tan'4 B 1 + tan 4 E 



- * 1 8fTtanT- 



2h cos' B 



(1 + tan* 4 E)< 



from whence the value of c might be found. 



Substituting in the equation (IV) the above value of dP, viz. 

 / dy \ dj: rf v 2t 



\ dJe) ~g' " d P uttin 8 tor 5^ >* value, j that equation 

 .become* 



* +*' 1 + t H _,/ 2t \ 



(l-t') + *'8- T^~t = * 2dx \"l^t>/> wn cncewe obtain 



t + t' 



(VI). 



But, as this expression does not admit of being integrated by any 

 known rules, mathematicians have endeavoured to obtain an aimroxi- 

 mate value of the integral ; and Bezout, whose method ha* been 

 adopted in th above Investigation, employs the following process for 



that purpose. Developing the expression t log. ~ in an infinite 

 erie*, putting that series in the form of a fraction, whom denominator 

 ^)*,and then ultituting it in Uw preceding equation, the 

 latter become., after reduction, 



(VII). 



Assuming the last {actor in the denominator to be constant, and 

 rnpinxintlng it by b, we have 



(VIII, 



which can easily be integrated, since the second number is equivalent 

 to the differential of a logarithm ; thus we have 

 2* 1 . 



where c* is a new arbitrary constant 



The value of b in equation (VII), when umjilificd, will be found to 

 be equal to 4 sec. K + ) cotan. E log. tan. (45+ 4 !.); and Bezout ha* 

 computed from this formula a table of it* values for every degree of 

 i-lt-v.ition. At 40" we have b= 1'1073 merely ; therefore, at elevation* 

 not exceeding that number of degrees, we may, without much vrrur, 

 consider b as constant and equal to unity. The value of c, when 

 obtained from equation (V) and simplified, is found to be equal to 



H 



4h coa L + b tan- E ' ^ *^ e Ti ' ue ^ ^ ""y ^* o **" 1 ^ by the 

 following process. 



2t 



At A, the point of projection, we have x o, and . _., = tan. E j 



therefore, at that point, the equation (IX) becomes 



1x 1 



= ^ log. (o-btan.E)-t-c'; 



whence 



m . Zx 1 4h cos E / 2bt \ 



Therefore = g log. -^ (c- j^,^ ; 



and putting e for the bane of the hyperbolic logarithms ( = 271828), 

 we have 



ill M* I 



(X), 



2t 



ml! -A 



2b. 

 B 



4h cos' i 



2bx 



odx H (/./ H 



y= ~ " 4bhcos'B 8 



This equation being integrated, and the constant determined on the 

 supposition that y = o when xo, we have 



2b_ 



OX H / H \ 



y= T + Sb'hcos'B V~ e )' or putting for c its value, 



[ 



ton. E + 



II 



4b*hcos* 



"I 

 iJ 



(i- ")..(). 



By substituting in this equation any assumed values of x, we should 

 obtain the corresponding vertical ordiuatcs ; and thus the form uf tin- 

 trajectory would be determined, approximative^. But, if the pbjeo 

 is merely to obtain the horizontal range, make y = o in the equation ; 

 then the latter will, after reduction, become 



2b r 2b /2bh \ -| 



x log. o = log. [I + ( sin 2 E + 1 ) .r J. 



SuK-iituting in this equation diflerent numbers for x, that which 

 renders the two members equal to each other will express the required 

 extent of the range. 



In tin- pivuoding investigation, g has been taken to represent the 

 accelerativc forco of gravity, or that by which the shot would descend 

 in vacuo ; but, in fact, it should represent the accelerativc force by 

 wliii-li the shot descends in air. And, in order to obtain the latter 

 force-, let r represent the semi-diameter of the shot, * tliu ratio of the 

 < in,, inference of a circle to its diameter ( = 8'1 4 1 59), D the density of 

 the air, anil it that of the shot. Then j) * r* D' will express the w.-ight 

 of the shot in vacuo, and J ir r 3 D the weight of an equal volume of air ; 

 *, } w r (D' B) is the weight of the shot in air, and J w r* g 

 (D'-rD) i tin- in. itivr |i--r by which the shot descends; the latter, 

 being divided by the weight of the shot, expressed as above, give* 



g for the accelerativo power required. But if the shot be of lead 



r irn, whose weight far exceeds that of an equal volume of air, the 

 term D may be considered a* equal to zero, and the accelerativc 



