330 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AMD H. W. RICHMOND 
The components of O are (0, 0, 6 \). Using the foregoing values of the components 
of A, X and O in equations (3.104) and (3.105), we obtain 
(3.201) N'=-Nr, 
as before ; the second and third components of (3.105) give 
(3.202) ANm' + B — In") — — J$mn6" l + —Bnld ,2 1 = n(zl—xn), 
(3.203) ANn' + B (lm"—ml") — 2Bnn'0\ — Bn 2 6 f ' 1 + B6"- l = ^{xm — yl). 
To solve the equations it is necessary to neglect certain terms. A discussion of 
the relative magnitude of the terms neglected, in various circumstances, will be given 
later in § 4.3. Some of these terms are negligible in all cases, on account of the 
smallness of 6 \ in comparison with the angular spin of the shell. Others are only 
negligible so long as the yaw S is so small that 1— cosd and 1—sin S/S may be 
neglected in comparison with unity. By such arguments it is not difficult to justify 
the reduction of these equations to the form 
(3.204) ANw'— Bn" + ANO'j = /u.(z—n), 
(3.205) ANA + BnU + Btf'j = n(m—y). 
For the particular case of the initial motion of the shells from the gun rifled 
1 turn in 30 calibres in the present trials, the terms neglected are, in general, less 
than 1 per cent, of some term retained, and the coefficients of equations (3.204) and 
(3.205) may be regarded as affected by possible 1 per cent, errors. Even in the case 
of the gun rifled 1 turn in 40 calibres, where values of S as great as 7 degrees or more 
are met with among the stable rounds, the employment of (3.204) and (3.205) is 
justifiable. 
We now define new variables and constants by the equations 
*i + c£ = m+in, c£=y + iz, 
AN/B = £2, c — cos 6 U 6\ + i6'\/Q — <f>. 
If we multiply (3.204) by i, and subtract from (3.205), we obtain 
(3.206) ^ + + 
So long as the yaw remains small, equations (3.201) and (3.206) may be taken as 
equivalent to (3.104) and (3.105). 
3.21. The Motion of the Centre of Gravity. —With the present axes, the 
components of G are (— ^sinfh, — g cos0 u 0). Equation (3.112) becomes! 
(3.211) v' = — B, (v, S)/m*—g (x sin 6 x -\-y cos 0j). 
t To avoid confusion the mass of the shell is temporarily denoted by in* 
