THE ROYAL ARTILLERY INSTITUTION. 
173 
Again, it is made still more convergent ; for equation ( e ) gives 
u x = + Aw a _ z .' §6. 
• — -^2 A ^Ux-l = — j2-A%a;_2Z 2Z 
X V A%*_z = + ^A%^ 2 Z 
— ^ A%*-z = — t¥o A 6 ^-2Z — Ho 1 tfux-u 
,\ by addition, and putting u x _$i = u x _ 3 i + A u x ^h &c., &c., 
I 2 C ^~ = A hi x .i — JgA%*_2z + JqA 6 «^3 z 
In order to apply these formulae (y) and (//'), 
Let 
&c. (h") 
((™ 33 
be the distance to any screen, and be denoted by “ s,” 
Let “l” be the distance between any two screens, 
Let u s be the time that the projectile takes in passing through the 
space “ s/’ and be denoted t s . 
Then A u s , ku a+z , &c., are the times occupied in passing through the 
successive intervals, and A u s -i is the time for the preceding interval^ 
and so on. 
A %u 8i &c., are the differences of these times. 
du^ = ~ = i, where v s is the velocity at the screen in question. (e)> 
cPs 
dt? 
( dt s \ s f 
-~j * = — from equations (c), where f s is § 2. 
ds ds 
dhi s __ dH s 
ds 3 ~~ ds 2 
the retardation (because negative) at the same screen. Hence we 
. obtain from (g) and (//'), by writing s for x and t for u, and making 
these substitutions, 
— .= At, — IA% + + &o. 
v s 
—• = A 2 t s —l — Yg-A^s —21 "h ‘yV^^s—3 1 ~ &C. 
V s = 
A t 8 — \£Pt s + JA 3 4 — JA 4 4 + &c. * 
/. = - (A 2 4-z ~ iV^- 2 l + t*A 6 4-3 1 - &c.) 
.( 1 ) 
.(*) 
Thus we have both the velocity and the retardation at any screen § 7* 
1=1 = 3,, oppose, 
dt 
d-t _dy __dy dt. dy 
ds 2 ~ ds ~ dt ' ds’ U dt 
m_ t (dt\ 
ds' 2 ~ dt 2 \ds' 
d~s 
dt 2 
as 
dt\ 
f Misprint here in Bashfortli—viz.. 
