TRANSACTIONS OP THE SECTIONS. 



subjects are considered, it is unnecessary to do more than insert the forms of them 

 which express the cu-ciimstances of om- problem, and explain the symbols em- 

 ployed. A pai-ticle is supposed to be projected with a given velocity (which in 

 the case of a falling particle may be zero) in a given dii'ection. The place on the 

 earth's surface, whence the particle is projected, is taken as the origin ; the axes of 

 X and of y are taken ia the horizontal plane, and are respectively north and south, 

 and east and west, the positive dii'ection of x beiag taken towards the south, and 

 that of y towards the west ; and the s-axis is the vertical liae measm-ed upwards 

 from the earth towards the zenith of the place ; and this liue may be assumed with- 

 out sensible error to pass through the earth's centre. The latitude of the place is 

 X ; and u> is the angular velocity of the earth ; g is the force of gi-avity of the earth, 

 and is considered to be constant for all points of the path of the particle (m). 

 Then the equations of motion are 



d-x 



de- 

 fy 



dt- 



—xa>^ (sin Xf — zap- sin X cos X+2 w sin X -f =0j 



'^=0. 1 



-yo)^— 2 a) (i 



siuX^+cosX 

 dt 



dt) 



dt 







^-x<B2 8inXcosX-sa)'^(co3X)=+2(»cosX^ 



=-.J 



Now » is a veiy small quantity ; to determiae its value I will take a second to be 

 the unit of time : then, as a mean sidereal day contains 86164'09 seconds, 



"=86^:09=13713= -O^^^^^. 



Consequently <b^, which enters into the preceding equations, is an extremely smaU 

 fraction. Also in the present problem, notwithstanding the increase of range now 

 obtained by the improved weapons of projection, x, y, z are all very small parts of 

 the eai-th's radius ; and therefore in the first approximate solution of the preceding 

 equations, I will neglect those terms which contain products of these coordinates 

 and of o)^ ; so that the equations become 





=0, 



d-y_ 



■2 



0) I si 



dx 



smX-^ + cosX^) = 0, 

 dt dtl ' 



^^+2coco9X t 

 df- dt 



= -9- 



As these are linear equations of the fii-st order, they are easily integrated ; and if 

 w=the velocity of projection, and a,, /3, y are the dii'ection-angles of the line of 

 projection, we have 



«=« < cos ee— M M sin X cos /3 i*, 



f 



y=ut cos ^+ua> (cos « sin X + cosy cos X) f-— w^cosX-, 



o 



B = M<COSy— (?+m<bcosXcoS|8| t^; 



which three equations give the place of the projectile at the time t. Now, without 

 proceeding further at present in the process of approximation, let us consider two 

 particular cases and resxdts, which are of consideraole interest. 



(1) Let the body fall, as e. g. down a mine, without any initial velocity ; then 

 w=Oj cos [«=cos/3=0j cosy=— 1; 



y= —a>gcos\ 





The first equation shows that there is no deviation in the line of the meridian : 

 from the second we infer a deviation towards the eastj that is, ia the direction 



