214 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
39. Formula for x 0 .—A formula lias been obtained in Article 32, and can be written 
in the form 
x«(r',s') = -5/if (/AAY|i + 3 0 fc=/Ak_A) + ...)?3(B' 1 -i + 2 By+ 3 By a 
/ [ (r +s ) cr J <x 
in which we have to put 
r' + s' = $2 (l 
(7 - S 2 {1 + (B'i + 1 ) ^+Bb<5 2 + B / 3 <i 3 + ...}, 
...) dS, 
r-r' = S 2 {B\S-e' + B' 2 $ 2 + B' 3 $ 3 + ...}, 
6 - / = mss'). 
After the integrations are performed the accents on S' and e are to be suppressed. The 
result may be recorded in the form:—The terms of the first order in the expression for 
T T*P 
-5h{l+e+$y{B\-l){S-S A ), 
the terms of the second order are 
-5h (1 +e + S) 5 {B'o-3 (B'i + 1) (B\-l)} {S 2 -S A 2 ), 
the terms of the third order are 
— 5h (1 + e + Sf { B' 3 - 2 (3 B\ + 1 ) B' a + 7 ( B\ + 1 ) 2 ( B\ — 1 )} (<f - Sf) 
-150/i (l+e + SY {±(B\-1) e-iB\(B\-l) (S + 2S A )} (S-S A ) 2 , 
and the terms of the fourth order are 
-5h(l+e + Sf {B' 4 —3 (2B'] +1) B'a—3 B'f + 2lB'i (B\+l)B' 2 
-14(B' 1 + 1) 3 (B' 1 -1)}(^-^ a 4 ) 
— 25A (l + e + (i) 4 [{2B , 2 —7 (BA +1) (B'l— 1)} e(3+2<S A ) 
{(3B'i — l) B' 2 —7Bd (B'j + l) (Bh — 1)} (^ 2 + 2d A <i + 3^ A 2 )] [S—Sff. 
40. State of the Gas at any Time .—With a view to applications it is important to 
indicate how the state of the gas may be determined at any time, or when the shot and 
its image have both travelled an assigned distance. We shall suppose that the time 
in question is an instant during the generation of the second reflected waves, before the 
second middle wave is obliterated. Then the central part of the tube is occupied by 
the second middle wave, and beyond-the junctions the rest of the tube, up to the shot 
and its image, are occupied by the second reflected waves. 
An assigned position of the shot and its image answer to a given value of x, and the 
corresponding value of S is to be found by solving the equation for x given in Article 34. 
