THE ROYAL ARTILLERY INSTITUTION. 
179 
where p and p' are the limits of integration; that is, this expression, 
if integrable, would give the time occupied in passing from an inclina¬ 
tion 0 to 6 r — i.e., from P to Q. Eq n (i). 
Evidently the time to Q is greater than that to P; that is, the time 
increases as tan <j> or p diminishes. This explains the negative sign. 
Again , 
_ dp dx 
^ dt dt 
dp dx j 2 
dx dt\ 
Eq« (1). 
U -l {l-y(8p +/)}-*; 
J 
« 0 2 p n' dp 
gj {1 -y (dp 
Again, 
dp 
dy dx 
dx dp 
u 0 2 PPP' pdp 
JJ {1-7(3? +/)}*’ 
(5) 
Equations (3), (4), (5), may be put differently. 
p — tan <p, dp = sec 3 <l>dip = (1 + p 3 ) d<j >; 
t 
«o (1 + p 3 ) 4 
g J {1 — y (3? +/)}! 
-2 y) suppose, ..,(6) 
9 
* 
(l + f) 
g J { 1 — y(3?+/)p 
n 0 3 4>rV 0° + jP 3 ) d(f> 
<J J { 1 — y(3?+? 3 )P 
0! 2 
= - (* J/) 
9 
u 2 
= - ^ (*J v +) 
9 
...(7) 
...( 8 ) 
A few words of explanation are here needed concerning the results 
obtained and the symbols used. 
“2b 33 is a variable, .\“y” is a variable, independent of 0 ; thus Eq n (m), 
the integrals cannot be solved except by giving “ y 33 every possible & § 9 * 
value in succession and treating it as a constant. Still, there are no 
known forms under which the integrals can be solved directly. This 
difficulty can always be got over by methods of approximation. By 
this means, Professor Bashforth has calculated tables ranging, more or App. ill. 
less, from 0 = 60° to 0 = — 60°, and from y = 0 to y = 5 (i.e., resistance IV * & VI * 
at vertex = 5 times the weight of the projectile). 
The symbols 1\ X, Y are merely abbreviations for the foregoing 
integrals; “ y” written beneath, denotes the particular value of a y 33 
to be used; and 0 , 0 ' are the limits of integration. 
* Misprints in Bashforth:— 
Last line but one in p. 52, “ dp ” should be omitted. 
Equation (7) denominator should be {l — y (Sp + 
24 
