Oct. 10, 1884.] 



KNOWLEDGE . 



201 



approximately, because, as compared with O M «, ab may 

 be neglecti'd. 



Now O J=2<, . V cos p OJ 



= 2<jn/i)- cos- t-|-it- + 2 It I' cos t . cos u 

 Hence we have 

 406. A 



a 6=- 



3 1 

 3 r(j 



v~ sin -£ V t;- cos -t -^ It" + i m t; cos t cos a 



Fig. ?. 



Now, in Fig. 3, let 01 ab represent the path O I ab of 

 Fig. 2, and let O //( X represent the latitude-parallel O mX 

 of Fig. 1. Put O k to represent the distance traversed by 

 O along this latitude parallel, in time 2<,,owing to the 

 earth's rotation, so that the point O is carried to k by the 

 time the missile has descended to a. Then k a is the real 

 range of the body ; and obviously k h is the range calculated 

 in the usual way, for in determining b we supposed the 



We have already obtained the value of ab / and we hire 



Im V cos £ sin a 

 tin ^ O ni = TTj = — T ^ ^ — - = — 



U I ^/■^•2 ^.Q^:: i^u-+2 UV cos £ COB « 



Om V cos £ CCS u-\-u 



CCS I O »i=— -T ^ — - — — -. — 



•J ' \ v- COS- ! + It- + 'I It V cos £ COS n 

 Hence we have, finally — 



■I'l ., . , 

 S. Dispt ^6 ?i^5 — V" sin- £ ccs t sin « 

 ^ 3r(/ 



8 

 = :;- <, /( u cos £ sin o 



W Di.-pt. =(i?i = .-T— (i;' sin- t cos £ cos n + »! '■• sin- t) 



= o~ 'i /< (« cos £ COS a -fw). 



A portion of the westerly displacement is indejitndnit i f 

 the angle of inclinaticn of the plane of flight lo the latitude- 

 parallel through the point of projection, and dejends only 

 on the time of flight, the elevation, and the latitude. TJih 

 last-named, we rememl>tr, is connected with it by tie 

 relation 



M = 2 TT r cos ,\-^no. of sees, in sidereal day. 



Substituting this value for jt, and putting P for the 

 number of seconds in a sidereal day, we have 



.nr i 1 1- 1 i <,/t /8i! cos £ cos a , 16-C0SX\ 



Westerly displacement =-1- i -)- ^ — I 



or the missile is projected vertically, we 

 16<,/t 



When £ = 90° 

 have 



horizontal velocity constant during the time of flight. 

 Thus, drawing a n perpendicular to b m, we see that the 

 point a actually reached by the missile lies south of b, 

 the point calculated in the usual way, by the distance b n, 

 and west of b \y the distance a n. 

 Thus we have 



Southerly displacement of ihe misbi'e = a i s-in /O?/; [ See 

 Westerly displacement =a 6 cos 10 m j Fig. 1 



Westerly displacement 



cos ,\, 



the value obtained in the first number of vol. IV. 



It will be observed that both in the expression for thp 

 southerly and westerly displacements, cos £ is always 

 positive; for £ is the angle of inclination with the hon- 

 zontal line, and is always le.^s than 90°. But a being the 

 angle of inclination with the latitude-parallel on the eastern 

 side of the point of projection, is measured (as angles aie 

 always measured) in the direction contrary to that in 

 which the hands of a watch move : this angle may have 

 any value from 0° to 3G0°. If « is greater than 180°, or 

 the projectile has a southerly direction, the expression for 

 the southerly displacement becomes negative, or the dis- 

 placement is northerly. In the southern hemisphere these 

 relations are net reversed, a projectOe directed nortl- 

 watds from a point in the southern hemisphere having 

 southerly displacement, while a projectile directed sonth- 

 waids has northerly displacement. A projectile directed 

 vertically has no displacement in latitude. 



If a is between 0"^ and 90°, or between 270° and 360°, in 

 othei- words, if the projectile is directed towards the east 

 (in either hemi.epheie), both terms of the westerly displace- 

 ment are positive. But if n is between 90° and 270°, or 

 the projectile is directed towards the west, the first term is 

 negative. The second term is always positive. In this 

 case the easterly displacement corresponding with the 

 negative first term may be equal to or greater than 

 the westerly displacement corresponding with the 

 positive second term. The relation which shculd hold for 

 these displacements to balance each other, is manifestly 

 obtained by equating the above expression for the westerly 

 displacement to zero, giving 



•lirr . 

 V cos t COSa =14= — -- cosX 



p 



If o^O, or the missile is projected at an angle i with tie 

 hcrizcn due west, we have 



rcos f=.U 



