52 
MINUTES OF PKOCEEDINGS OF 
in mathematical calculations, whenever the positions of several points on a 
plane surface have to be considered, and called the method of rectangular 
co-ordinates , the application of which to the case in question is exceedingly 
simple and easy. 
17. Having found the probable mean centre of impact as before (by 
taking the centre of gravity of the whole of the shots registered) draw 
through it a vertical and a horizontal line, as in the following figure. 
Fig. 4. 
Then, taking any shot a as an example, its position will be accurately 
defined by two co-ordinates, x and y, drawn parallel respectively to the 
horizontal and vertical lines before mentioned. We will call the distance y 3 
the vertical error of that shot, and the distance x 3 its horizontal error. 
18. Suppose next we wish to find the mean errors in both directions. 
The most simple and natural mode is to add all the vertical errors together, 
(both above and below, without regard to sign) and to divide by the total 
number. This will give the mean vertical error; the mean horizontal error 
may be obtained in the same way. These mean errors are said to be 
determined lineally. They correspond with the <c ecart moyen 33 of the French 
artillerists, and we may denote them, in the French notation, by the symbols 
XandZT respectively. So that 
IS x 
II = mean horizontal error (determined lineally) = — L - 
(3) 
K = mean vertical error (determined lineally) 
( 4 ) 
19. But the term mean error is somewhat vague. A mean quantity is 
not necessarily a simple arithmetical mean; for the way it is advisable to find 
it often depends on the purpose for which it is to be used. In this case, 
the whole problem being one of probabilities, mathematical considerations 
teach us that we shall have a greater chance of being correct, if we estimate 
the mean error in another way, viz. by adding together the squares of the 
separate errors, dividing by the number, and taking the square root of 
the quotient. The mean errors thus obtained are said to be determined 
