Mr. J. J. Sylvester on Projectiles, 451 



velocity V, supposed to make an angle 6 with the horizon, as 

 also g, the accelerating force of gravity, each into two parts, V 

 into V cos {6 + i) and V sin {6-\-i), and g into g sin i and g cos i, 

 respectively parallel and perpendicular to the plane of the slope. 

 The time of flight is of course found by looking to the per- 

 pendicular part of the velocity and of gravity alone, and is evi- 

 dently 2 ^^^^ S which call t ; the range will evidently be 

 •' g cos I 



X^Sll^ ,-. ,. _Z!_/sin {20 + i) + sin i\ 

 cos 2 gcost^ -^ 



Hence the best angle of range for this case is found by making 

 20 + i=9O,e=^{9O-i). 



Now let us proceed to apply this result to the general case, as 

 in the figure below, where BC is the slope upon which the range 

 is to be measured, A the point of 

 projection, AD the direction which 

 gives the maximum range upon 

 the slope, and BC the aictual ex- 

 tent of this range ; then I say AD 

 is the direction which would give 

 also the best range upon the slope 

 AC. Since if, with the given ve- 

 locity of projection, any other di- 

 rection than AD would give a 

 better range upon AC, the path corresponding to such direction 

 must evidently cut BC at a point beyond C in that line in order 

 to strike a point beyond C in the line AC. 



Hence if we draw the horizontal line AE, we know by the 



preceding case that the angle DAE = — CAB*. 



Let CAB = ^, which is to be found; also let AB=h, and the 

 inclination of BC to AD = t, h and t being given ; and let t = 

 time of flight, then 



CAD = (90°-(^) + | 



(=90»-|). 



* This equation, and the isoscelism of the principal triangle of the figui-e 

 to which it leads, would not readily present themselves to notice in the 

 direct method of seeking the maximum range. It is for the sake of this 

 pleasing geometrical relation, not unmixed perhaps with a desii-e of ex- 

 hibiting the simple yet dehcate turn of reasoning, the agreeable little point 

 of method (a fly embalmed in amber) contained in the immediately pre- 

 ceding paragraph, that I have thought this trifle worth iireserving in the 

 pages of the Magazine. 



