The theoretical treatment of the subject, however, affords a safe guide in the process of solving the inverse problem and gives a clear indication of the form of the equation of the trajectory. Upon supposing that there is no air resistance a limiting form is secured, affording a minimum number of terms of the expression for y in terms of x. This is the condition in vacuo.
The equation immediately takes the form gx2 Y'x tan # 2 In cos' and 2gx y-..= tan 0 tan 2 vicose 2g Yr= 2 VI COS* 0 for this condition.
From the equations of motion in air it is clear that any one of the variables x, y, v, may be expressed in terms of any other of them and hence every one of these may he expressed in terms of x. As complete integration is not always feasible, it follows that in the general case the variable in question will be expressed in terms of ascending powers of x. The equa tion of the trajectory, then, takes the general form 1 + ax +etcl x tall. 2 Mos' # tart # 2 ax +The+ etc} 2 dly 1 +3ax +IV& + etc } 2 Vico& # .
2g 1 3a + 12 bx+etc 2 cos' whence 2gx tan 0= tan 0 2 v.j cos2 3 1 + ax + 2 cos' 1 vs cos* 6 3a + 12 bx+ etc 1 By this means a set of actual firings may be used to determine the law of motion of the projectile since the muzzle velocity, V, the angle of departure, 0, and the values of y and x for the point of impact are measured.
The value of F(v) is readily placed in the form F (v) 3 cos' -1 a +4bx +etcl. cos 0 C 2 k Vicoss# 3 d +4bx +etc r cos 8 - 2 a +34x +etc ) Now, cos 0 is readily expressed in terms of x, since tan 0 is so expressed, and 1 cos 0 =- V1+ tan' 0 v) Hence is directly expressed as a function vi of x.
When x becomes zero 7.11, and 0=0, and hence 3 FV 2 a C see # or 2 F(V) cost 3 C Hence 1+4 x +etc F (v) a cos e. cos # -- v' F (V) 1 +3ax On taking successive derivatives with respect to x, and noting the relations of v and 0 to x, the values of b and subsequent coefficients are determined in terms of F(V), V, and the suc cessive derivatives of F(V) with respect to V. It is quite clear that Q will invariably enter each value. This method is general, and finds useful application both in the case of high power guns with small values of ’ and in the case of howitzers and mortars, since the ve locity v, the angle 0, and all other elements may be directly found. The value of t is given by the relation, Vt cost= V (1 a Whence Vt 3 =1+ 4 ax +( 3 b +etc 8 Where the value of # is large the solution of the inverse problem is practically essential, in the light of recent ballistic firings. Where
the angle # is less than 15°, however, a method due to the eminent Italian authority, Siacci, greatly simplifies the problem and permits very satisfactory discussion of the trajectories of direct-fire guns. An auxiliary variable known as the is characteristic of his method. Its value is given by the equation u =-- v eos 0 sec In connection with this he assumes the relation F (v) = p F (u) cross sec 0 in which p is a quantity which, for direct fire, is very nearly unity. Its values are usually tabulated with the values of Q and X as arguments.
Wherever Siacci's method is used the value of C will be understood to include the factor I in its denominator. With these modifications the following relations are found: du Coss d tan 0 u F (u) = gC de C sec # u (u) ud dx= Cu) F (city= F (u (tan 2 cos' 0' iI(u)-I(V) 1 in which I 2 gdu uF (u) Placing S (u)=f u udu F (u) T u du F (u) A (u)=f I (u) d S (u) the solution of the problem appears in the equations C' J A (u)A (V) y=x tan # (u)S ( / ( V) tame= tan 2' (u) 1 (V) C sec Q f T (u) T (V)t x (u) S (V) It is clear from the above that the value of 1 F(u) is more convenient than that of F(u). Such a Nalue may be found in ascending powers of u beginning with a term involving or in other u words F (u) may be determined by a series the first term of which is a constant; and the other terms involve regularly ascending powers of is. With complete and absolutely satisfactory data the curve of these values may be found and the direct problem for direct fire is completely solved when # is lcnown.
One of the most recent available experimen tal determinations of F (v) is that of the Gfivre Commission, and certain broad characteristics are noted. They are: 1. Between zero and 800 feet per second F (v) is roughly proportional to vz.
2. Between 800 and 1,600 feet per second it is roughly proportional to v`.
3. Between 1,600 and 3,600 feet per second it is roughly proportional to v%.
These three classes correspond respectively to the fire of 1. Howitzers and mortars 2. Field guns 3. Seacoast guns In these three cases the character of F (v) is such that it may be represented, for certain purposes, by F in which B and n are constants. In such cases the equation of Siacci's trajectory is gx2 y x tan 0-- I +ax -1-bx2 cri +tic 2 Incos2 so in which 2 B V a 0 3 C 4-n /.8 V n_2)2 6 C .(4-n) (6-2n) (B 3 C 30 etc.
4 eFor the values n=4- and equations take the simple forms: 1. For n=4 gx22 B V2 y = x tan 0 (I 2 V2 cos2 3 C 4 2. For n= gx2 2 B V-21'-2