TRANSACTIONS OF THE SECTIONS. » 



We shall suppose the values of x, y, z mven ahove to be approximate solutions 

 of the first order of the equations; and if, according to the general method of 

 solution adopted in such cases, we substitute these values in terms involving 

 a>^x, (o^i/, and (o-z, that is in the smallest terms which we intend to retain, and, 

 omitting terms of higher orders, then integrate the simultaneous ditferential equa- 

 tions thus formed, the results are 



x=ut cos x — uay siuX cos/3 f^ 



— u ar^ sin X (cos ot sm X+cos y cos X) ^+gaP' sin X cos X — ; 



J 8 



y=u t cos /34-» <a (cos a sin X+cos y cos X) f^ 



— ca g coa\ ——u (d^ COS ^ —; 



3 "^2' 



z=u t casy—^g t^ — ua cos X cos /3 1"^ 



— M o)^ cos X (cos 06 sm X 4-cos y cos X) ^ +^ w^ (cos X)^ — . 



These equations, of course, give results corresponding to particular initial circum- 

 stances. I will take only two. 



(1) Let the body fall without any initial velocity; then m=:0, cos «=cos /3=0, 

 cos y= — 1 ; 



a;= 0)^ (/ sm X cos X ^, 



^3 



?/=-a)^C0sX-, 



> 



s=-|^<^+o,V(cosX)^| I 



The first equation shows that there is a deviation of the falling particle in the line 

 of the meridian towards the south ; and the second shows that the deviation in the 

 parallel of latitude is towards the east ; so that the resulting deviation of the faUino- 

 body is towards the south-east. This result is in accordance with the case many 

 years ago investigated by Hooke, the contemporary of Su- 1. Newton. From the 

 last equation it appears that the space due to a given time is less than it would be 

 if there were no rotation. 



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

 6, so that cos «= cos 6 ; cos ;3^0 ; cos y=sin 6 ; then 



a:=M<cos5— M cousin X sin (X -I- ^} --\-g ai^sm. X cosX- 

 y=M<Bsin (X+^) i!--a)^cosX|, y 



z=M«sinfl-^-Mo)2co8 X sin (X+^) ^+^0)2 (cosX)2 ^. 



ii 2 8 



2 1< sin ^ 



\Mien the projectile strikes the ground, 2=0; and approximately t= ^^'^^^. in 



which case y=i^_(^5i)!{sin 5 cos X+3 cos 5 sin X } ; 



which is the same expression as that just now intei-preted. Consequently the aim 

 of a long-range gim pointed due north or south must be in accordance" with the 

 preceding explanations. 



On the Calculus of Functions, with Remarks on the Tlieory of Electricity. 



By W. H. L. Russell, A.B. 

 The object of this paper was to give some account of a method discovered by the 

 author for the solution of fimctional equations with rational quantities, known 

 fimctions of the independent variable, as the ai-guments of the imknown fimctions. 

 The solutions were given by series, and also in tenns of definite integrals. 



