114 
CAPTAIN NOBLE AND MR. E. A. ABEL ON FIRED GUNPOWDER. 
If, for example, we endeavoured to represent the relation between s and t by a linear 
equation of the form 
s = cut -J- bt 2 -f~ ct 3 4“ dt^ &c., (8) 
we should have to determine the most probable values of the coefficients a, b , c, d, See. 
from the eighteen to twenty direct observations connecting s and t. The equation 
would further have to be such that the first and second derived functions should repre- 
sent curves of the general nature we have described. It is obvious that, setting all 
other considerations aside, the labour of such a series of calculations would be insur- 
mountable. 
But although it is impossible to obtain a single relation between s and^ t for the 
whole length of the bore, we have endeavoured, on account of the great importance of 
the question, to obtain such a relation for the commencement of motion, where the 
question of pressure is of vital importance. 
To do this we have taken only the observed values of s and t so far that we could 
be certain the position where the maximum pressure was attained was included, but 
have made no assumption whatever as to the actual position of maximum pressure. 
We then assumed that the relation between s and t was capable of being expressed 
by an equation of the form 
s=at a+pt+yt2 ; (9) 
and from the observed values of s and t the probable values of a, a, (3, y were com- 
puted by the method of least squares. 
Treating in this manner the experiment under discussion, and taking from Table IX. 
the first six values of s and t *, we have obtained for the most probable values of 
o= 3-31076, 
«= 1-37851, 
(3= -76600, 
y= — -06932, 
and for the relation between s and t the equation 
5=3-31076 1-37851 + V6600i- -06932*2 (IQ) 
By differentiation we obtain for the velocity 
ds s 
^=V=-{(a + /3^ + ^ 2 ).(l + log^)-(«-yf).log^}; (11) 
and by a second differentiation, 
rp W d 2 S 
pressure T=-.^ 
+ r^X 1 + l °ge t) — (aL— yf)log e t - 1 } 
-f.^{/3-fO-4y^).( 1 + log e ^} (12) 
* For the convenience of calculation, the unit of time used is not a second hut the thousandth part of a 
second. The unit of space is altered in like proportion. 
