On a Construction for the Range of Projectiles, 113 



The point of projection and point of impact speak for them- 

 selves ; the point vertically over the point of impact in the di- 

 rection of projection may be called the point of aim. The line 

 joining the point of aim and the point of impact is the drop or 

 fall', the line joining the point of projection and the point of 

 impact may be called the excursion ; and that joining the point 

 of projection and the point of aim, the length of aim, 



A vertical section of the ground (plane or curved) through 

 the axis of the gun may be called the field. We may then say, 

 that, for the maximum range, the fall is always equal to the 

 excursion, whatever the form of the field ; and that in general 

 the locus of the point of aim, for a rectilinear field when the 

 point of the projection and the velocity are given, is a circle to 

 which, in the case of the angle of best elevation, the line of fall 

 is of course a tangent. It would not be surprising if a good 

 deal of elegant geometry (like ivy twining round an old wall) 

 should hereafter associate itself with Mr. Galbraith's " circle of 

 aim ; '^ a propos of projectiles, it is not unworthy of observation, 

 that the velocities at any two points P and Q of the parabolic 

 path are as the lines PT, QT which the tangents at P and Q mu- 

 tually cut ofi" from one another, a remark which of course is easily 

 seen to extend itself to the case of an elliptic orbit with the force 

 in the centre. 



Ever, Gentlemen, 



Your faithful friend and reader, 



Woolwich Common, J, J. SYLVESTER. 



July 3, 1856. 



P.S. The value of Mr. Galbraith's method consists simply in 

 the act of conception of the locus of the point of aim ; it was 

 scarcely worth while (at this time of day) to append a synthetical 

 proof of so simple a proposition, which may be got at immedi- 

 ately by calling the length of aim p, its inclination to the ver- 

 tical, 6y and that of the field to the vertical, i ; when by similar 



triangles (if H denote the quantity — , and tj the vertical distance 



of the point of projection from the field) we obtain the equation 



H "^ ^ 8m{i-e) 

 p sin i ^ 



or 



'^ smz ^ ' * 



which obviously corresponds to the circle of Professor Galbraith. 

 Phil. Mag. S. 4. Vol. 12. No. 77. Aug. 1856. I 



