ON LAGRANGE’S BALLISTIC PROBLEM. 
215 
This is the value of 8 at the image of the shot, and the corresponding values of r and .s 
at the image of the shot are given by formulae in the same article. Also, 8 being known 
for the image of the shot in this position, the value of t is given by the formula of Article 
33. Let this particular value of t be denoted by T 3 , and in like manner let the values 
of the various quantities at the image of the shot at this time be denoted by attaching 
a suffix 3 to the letters, thus :—R 3 , 8 a . 
In the second reflected wave from the left the values of r that occur lie between R, 
and R 3 . To each such value, when t = T 3 , there answers a value of s and therefore of 
8. If in the formula of Article 37 we put T 3 for t and the chosen value for r, the formula 
becomes an equation giving 8. The chosen value of r determines the corresponding 
values of e and o A , and the deduced value of 8 determines the corresponding value of s. 
Then, simultaneous values of r and s being known, all the quantities can be determined. 
It seems to be most appropriate to assume a series of suitable values of r and calculate 
the corresponding values of s. The process of finding 8, by trial, may be simplified by 
means of a theorem to the effect that the loci, in the plane of (r, s), which answer to 
constant values of t and x 0 , are equally inclined to the axis of r. To prove this we 
have 
du 
dc , ' t = const. 
dt jdt 
0 c t du 
dXo 
fdx o 
0 ^ 7 / 
0CT 
do- \ 
du) xo = const. 
or 
/ dr — ds \ _ l dr + ds \ 
\d'V + ds It — const. \dr dsjxo = const. 
or 
dv)t = const. 
+ 
xo = const. 
= 0 . 
This theorem shows that a point of given r on the locus t = T a is not far from the image 
in r = R 3 of the tangent at (R 3 , S 3 ) to the locus, along which x 0 = 0. Hence a first 
approximation to the s answering to a given r is 2S 3 — s A , where s x depends upon r in 
the known way, and therefore a first approximation to the required value of 8 is 
2 S a -S A . 
The junction of the second reflected wave from the left and the second middle wave 
is characterized by the value R 3 of r. If, then, the process indicated above is carried 
out for the value R 3 of r, the result is to give a pair of simultaneous values of r and s, 
which can occur in the second middle wave at the time when t — T 3 . Another pair of 
simultaneous values can be found by finding the common value of r and s which occurs 
at the central section at the same time. This is to be done by putting r = s and t — T 3 
in the formula giving t in the second middle wave, and solving the resulting equation 
for r by trial. When this is done we shall have two pairs of simultaneous values of r 
and s which occur in the left-hand half of the central part of the tube at time T 3 , and 
they are the extreme values of r and s which can occur in that part at that time. To 
obtain other pairs, we may choose an intermediate value of r, substitute in the equation 
giving t the value T 3 of t and this value of r, and find s by trial. For a first approximation 
2 H 
VOL. CCXXII.-A. 
