A CALCULATION OF PROBABILITY OF FIRE. 57 
(4) By referring these results to a probability table to find the 
percentage of shots that should fall in the length and in the breadth of 
the objective thus ‘ projected’ on to the horizontal plane. 
(5) From this to calculate the probable percentage of effective 
shots. 
The area of the horizontal equivalent of the leLiyet presented by a 
ship will depend on 
(1) length, breadth, freeboard of ship ; 
(2) gun used ; 
(3) height of gun above sea level ; 
(4) range ; 
(5) angle of approach. 
The outline of the ship may be considered to be an ellipse whose 
axes are the length and breadth of the ship. 
Let acbd be such outline, let LZ and / the axes of this ellipse be the 
length and breadth of the ship, h its freeboard. 
@ the angle of arrival of the projectile, i.e. angle of descent for range 
ap depression angle. 
¢ the angle which the line of fire mn, makes with a, b the length of 
the ship. 
eae be another outline of the ship parallel to the first at a 
distance mm, dependent on @ and h. 
Draw pp,, 77, tangents to the two ellipses; the area equivalent to 
the target is then the space pmgqq,n,p,, and this space may be con- 
sidered equal to a rectangle whose length A is mn,, and whose breadth 
B is such that the area of the rectangle will be equal to the area 
UO GI a 
The rectangle AB can be calculated from the properties of the 
ellipse. 
Lxt h 
=r Set adiootioudod afouduosudaduodu iD), 
VE cos? fb + L? sin? p tan 6 (1) 
7 h Poot t Tae o 
Ec ew ¢+L7 sin? & 
eens ey | an eal Deere ero eereerecsssoes ° 
Now supposing the ship were stationary and the firing carried out 
with the same accuracy as when firing to obtain the length M and 
breadth NV of the 50 per cent. zones, the percentage of effective shots 
could be found by working out the values of a and. of 4 , and referring 
them to a probability table such as is found in the Appendix to 
M°Kinlay and multiplying the two percentages together; thus, if 
1 4 (taken as 100 per cent.) and re 1 or 50 per cent., the percentage 
of hits would be 50 per cent., or, if the value of s gave 75 per cent. 
