^2 



,3. 



y= ^r— ^s% 



6 REPORT — 1861. 



towards which the earth is moving ; which varies as the cuhe of the time of falling; 

 iand that this deviation is gi-eatest at the equator, where \=0 : and the last equa- 

 tion shows that the earth's rotation does not produce any alteration in the time of 

 falling. 

 If we eliminate t, and take z downwards to be positive, 



■_(cosX)2 



which is the equation to a semicubical parabola, and shows that the square of the 

 deviation towards the east varies as the cube of the space through which the par- 

 ticle has fallen. 



(2) Let the particle be projected due southwards at an angle of elevation equal 

 to 6; then 



cos a8=cos 6, cos /3=0, cos •y=sin 6; 



and x=ut cos 6, ~\ 



f3 1 



y=UKOsm(d+\) t^ — cogcosX —, [ 



z=ut am 6—?--. j 



From the first and the last of these equations we infer that neither the time nor the 

 range on the meridian is altered by the rotation of the earth. Also when 3=0, 



that is, when the projectile strikes the groimd, <= _?i^l5_; in which case 



9 



4 M^ £0 (sin &Y t • n X . o /I • \ 1 



y= -^ — ^/sm^cosX + Scos^smX}; 



6<j- 



and therefoi'e the point where the projectile sti-ikes the groimd is west of the meri- 

 dian so long as ^ is less than 180°— tan ~^ (3tanX): and the deviation vanishes 

 if ^=180° — 3 tan~' (3 tan X). The deviation is eastwards if d is greater than 

 180° -3 tan ' (3tanX). 



Now these results, which have herein been applied to the motion of a material 

 particle, are also true of that of the centre of gi-avity of a body. Neglecting there- 

 fore the resistance of the air, and the action due to the -rotation of a ball or bolt, 

 they are applicable to rifle and cannon practice, and we have the following results. 



When the shot is fired due north or south, the range in that direction is not 

 altered, but there is always a deviation of the shot, the value of which at the point 

 of impact on the ground is given in the last equation. 



Also from the preceding equations the following residts may be deduced : — 



When the shot is fired due east, the range eastwards is increased or diminished 

 according as the angle of elevation of the gim is less than or gTeater than 60° ; and 

 the deviation is southwards for all places in the northern hemisphere, and north- 

 wai'ds for all places in the southern hemisphere. 



When the shot is fired due west, the range is increased or diminished accoi-ding 

 as the angle of elevation is gi'eater than or less than 60° ; and the de-viation is 

 northwards for all places in the northern hemisphere, and southwards for all places 

 in the southern hemisphere. 



So that for firing fi-om a place in a direction coincident with the parallel of lati- 

 tude, and with an elevation less than 60°, the range is increased or diminished 

 according as we fire eastwards or westwards ; and the difierence between the two 

 rano;es 



'"^{3 (cos 5)2 -(sin (9)^}; 



3^ 



and if the place is in the northern hemisphere, the deviation parallel to the meri- 

 dian is north or south, according as we fire west or east. 



Now these eflects have been inferred from the equations of motion, simplified by 

 the assumption that products of oj-, and one of the relative coordinates of in, are 

 small quantities, and are to be neglected. Let us now retain these quantities, and 

 assume that products of co^ and of a small variable are to be neglected ; and that all 

 small quantities of a lower order are to be retained. 



