258 



NATURE 



\yan. 1 6, 1890 



ON CERTAIN APPROXIMA TE FORMULAE FOR 

 CALCULATING THE TRAJECTORIES OF 

 SHOT. 



TN the postscript to a paper by Mr. W. D. Niven, "On the 

 •*■ Calculation of the Trajectories of Shot," which is published 

 in the Proceedings of the Royal Society, vol. xxvi. pp. 268-287, 

 I have given, without demonstration, some convenient and not 

 inelegant formulas applicable to a limited arc of a trajectory 

 when the resistance is supposed to vary as the «th power of the 

 velocity. 



In these formulee, the angle between the chord of the arc and 

 the tangent at any point is supposed to be always small. The 

 index n is not restricted to integral values, but may take any 

 value whatever. 



As the proof of these formulae is not altogether obvious, and 

 a similar method of treatment may be found useful in other 

 problems, I think it may not be unacceptable to your readers if 

 I show here how the formulae may be demonstrated. 



Analysis. 



Investigation of formulae applicable to a small arc of a 

 trajectory, when the resistance varies as the wth power of the 

 velocity. 



Let X and y denote the horizontal and vertical co-ordinates at 

 time /, u the horizontal velocity, and ^ the angle which the 

 direction of motion makes with the horizon at the same time. 



Hence the velocity at time ^ is « sec 0, and we may denote 

 the resistance by kti"(%QQ. <^Y, where k is constant throughout the 

 small arc in question. 



Also let/ and q denote the values of u at the beginning and 

 end of the arc, o and 6 the corresponding values of 4>, g the force 

 of gravity, T the time taken to describe the arc, X and Y the 

 corresponding total horizontal and vertical motion. 



Making <^ the independent variable, the fundamental formulae 

 are — 



(4) ^i = - -(sec <pf. 

 d<p g 



From the first of these equations — 



I du k, ^s„ , 1 

 — — — , -,- = -(sec d>)« + 1 ; 



and therefore, by integration between the limits (p = a and 



i I _ kn f" 



r ~ P" ~ JJ (sec<^)«V^«/'- 

 Also, we have — 



X = 



and 



I / o 



- / «-(sec <p)-d<b ; 

 gj y9 



Y = - / ti-(%&c 0)'- tan <p d<f> ; 

 SJ P 



T = - / ti{sec <f>y-d(b ; 

 gJ fi 



and we wish to compare the two former of these definite integrals 

 with the following known one, viz. : — 



and the last with — 



r 



p" 



- = (« - 2) 



J fiU"-^ 



^"d<t> = '^ 



2) 



'i: 



u^sec <p)" + ^d(p ; 



, = (« - l) / — -,-d(p 



k{n - I) 



«(sec </))" + V^). 



This may be done by means of the following lemma, which follows immediately from Taylor's theorem : — 



Lemma. 



If F((^) be any function either of ^ only, or of <^ and ti, where I and if o and /3 be the limiting values of <p in the integral and 

 n is a function of (^ given by the above differential equation (i), | 7 = A(o + j8), then, putting for a moment (^ = 7 + co, 



rF(<^)^<^= f ^'''"^'F(7 + ccKa, = f *'""''' I F(7) + F'(7)« + F"(7)-'*'' + F"'(7)^' + F""(7)''~ + &c. U/co 

 ) B J -A(a-^) J ~iU-^>^ 2 6 24 J 



= (a-;3) j F(7) + i-(a - &fF"{y) + ^y {a - fi)*F""(y) + &c. | 

 I 24 1920 J 



where F'(<|>) = ^\ F"(^) = ^^Uc, and F(7), F'(7), 



F"{y), Sec, are what F{<t>), F'(4>), F"((/)), &c., become when 7 is 

 substituted for f, and the corresponding value of u {iiq suppose) 

 is put for ti. 



In what follows, the last of the terms above written, which is 

 of the 5 th order in (a - j8), is neglected, together with all terms 

 of the same order of small quantities. 



All the definite integrals with which we are here concerned 

 are included in the two forms 



/ tc^(stc <p)"'dtp, and / u'{sec </>)'" tan <p d<p. 



J /8 J ff 



j (sec (p)" + ^d<p = (a - /8)(sec 7)'' + 1) | 

 Hence 



I + (a 



24 



In the first place, we will apply the above formula to the 

 case in which ¥{<{)) is a function of <{> only, viz. when F(<^) = 

 (sec (p)"- + 1. 



Hence 



F'{<p) = (« + i)(sec </>)"+! tan <p ; 



¥"{<!>) = (u + !)[(« + i)(sec </))«+i(tan <p)^ + (sec <p)" + '^] 



= (« + i)[u~+~2{sec <j>)" + -^ - n -r~i(sec <^)« + ^]; 



and therefore, 



)3)2 [« + 2(sec7)-- n + i] }-, to the 4th order inclusive. 



- - -- = ^^'(a - m{sec 7)" + 1 1 1 + "±^(a - ^f\n + 2(sec 7)2 - « + i]|, 



P" g \ 2\ J 



or 



F'(^) = F{<p)r^~u"{sec <[>)" + '^+ i'«tan<^~|; 



p 



■which gives g when / is known. 



In the next place, let F(<])) — u\sec (p)"'. 

 Hence 



F'{<p) = ^ = lu' - ^~ (sec ^)'« + mu\sQc <p)'" tan <p 

 j<p d<p 



