GEOMETRY. 



To det^nnine the relation between th; elements of pofition 

 a, b, a, i ; <v, $, a', /3', eliminate r,j(, z, from the fourcquations, 

 ind (a' - a) {b' - b) - {l3' - 13) (x — a.) = o. 



Of the Equation to a Plane. — A piano being given by its in- 

 terfeftioii with two co-ordinate planes, it may be conceived 

 ns generated by one of thefe lines moving parallel to itfelf in 

 the direclion of t'le other. 



Let z = ax + c, and z := by + c, he the equations to 

 the two given interfcfting lines, tiie generating line being 

 parallel to itfelf, and to the inti-rfccling line on the plane 

 X z, its equations in any one pofition will be 



z = ax + y, y — p. 

 But if it (lioiild niL-ut the fecond line, wliofe equations are 

 X — o 



z ^ by + e; 

 hence this e<]uation of condition 



ip + c = y; 

 from which it follows, that the equation of the generating 

 line of the plane, in any given pofition, depending on jS, are 

 z=ax-\-bi3-i-c 

 y = li. 

 Eliminating p from thefe two equations, that of tTie plane 

 is obtained, 



2 = rt X + iji 4- f . 

 in which a and b are the tangents of the angles which the 

 interfcftiona of the j)lan£ make with the axes .v and y;c\% the 

 co-ordinate z, correfponding to the erigin of the ordinates ; 

 lince, if in the equation of the plane, x and_y are made =; o, 

 ■z r= c. This equation may be more commodioudy exprelfed 

 by the following form : 



A K + Bj- + C s + D = o, 

 in which, of the four conllant quantities A, B, C, D, three 



\~r' 



only are neceffary ; hence <( — 7^ ^^ ^ 



D 



The equation of a plane has been determined from its inter- 

 feftions with the planes of the co-ordinates. Thefe interfec- 

 tions may llkewife be determinedby the equation to theplane. 



Let ax -j- by ■{- c = z, make fuccelTively x =■ o, j/ = O, 



z =. o, and there refult < , t i ~ ^ , 



\ax-\-by-\-c^:=^oiz = ax-\-c 



■J ■' 3 . which equations belong to the interfeftions 



of th? given plane with the three planes x y, x s, and xy. 



The equation of a flraight line, iituated in one of the 

 co-ordinate planes, is likewife that of a plane paffing through 

 this line, and perpendicular to the plane of the co-ordinates 

 which contains it. "When the plane is perpendicular to one 

 of the axes, as x, its equation is .v ;= c (conilant) : _y =; /9, 

 i = y, are the equations of two other planes, one perpendi- 

 cular to the axis •/, and the other perpendicular to the axis z. 



In the equation to the plane make fuccelfively -j ~ J- , 



•}^~f> •[■''ll'^f; values will then be obtained for 



X, y, z; which are the dlflances of the origin of the co-or. 

 dinales from the points of interfeclion of the plane with the 

 axes of the co-ordinates : let the equation to the planes be 



c c 



%=^az-\-bf-Y c, thefe diftances will be — — , r ' ^* 



a b 



Two planes which are parallel have parallel intcrfcAions ; 

 therefore, if the equation to the firft plane be 

 Vox.. XVI. 



t.r=aii:-\-ly^e 

 the fecond, % r=. a x -Y b' y ■\- c' 

 the condition of parallelifm will be exprcfled by die equar 

 tions a = a', b = i'. 



Pkoblem L 

 Problems relating to a Jlraighl Line and a Plane. — To 

 draw a plane parallel to a given plane, let the equation to At 

 given plane be 



z-=^ax-rby-\-d 



and that of the plane required 



2 = a' x -I- by -\- d 



then the condition of parallelifm will be 



n' = a, b =1 b. 



PROnl.K.M H. 



To determine the equation of a plane which (hall pafi 



through three given points, let the co-ordinates of the 



given points be i .v',_y', z', 



2 •v",y,z". 



1 r'" v'" •!■'" 



The equation of the plane required being fnppofed 

 Ax + By + Cz + D^o 



the three following conditions are obtained ; 

 A x' + By' + C 2' H- D = O 

 Ax" + By" + Cz" +D = o 

 A a"' -f B^'" 4- C ='" -1- D = o . 



from which are deduced the following equations ; 



A-y (a'-2"') +j,"(2'"_2') ^yU'-z") 



B z= 2' (x" - x'") + z" {x'" - .v') + z'" {x' - x") 



c ^ x' (/ -y") -t x'" {y"> -y) + X'" (y - y) 



D =x'(y" z"'-y'" z") + x' (y a'-y' z'") + x'" {y z"-y"x') 



The three co-efficients to determine are js > v, » ;7 from the 



fame number of equations. If tlie triangle formed by the 

 flraight lines joining the given points be projected on the 

 three planes zy, x z,yx, the areas of thefe refpe£live pro- 



A B C 



jeftions will be — , — , -,and it will be (hewn that D is fix 

 •' 2 2 2 



times the folidity of a pyramid whofe bafe is the triangle in 



fpace, and whofe vertex is the origin of the co-ordinates. 



Let a triangle, as z' 2" 2'", reprefent the projections of 



the above three points on the plane of « 2, the area of the 



tx"i x") z{x"' — m''\ 



trapezium z" Ji'' x'" z'" will be -^ -' -f- — ^^ : 



2 2 



the area of the trapezium x' 2' 2" x" will be 

 x" jx' - x') z[Jx"' - x") 



+ 



2 2 



»nd that of the trapezium x' 2' z'" 

 z"'{x"-x ') , (z'a- 



2 



-- +■ 



*'" will be 

 ■■■<') 



From the fum of the two firft furfaces, take the laft, the 



difference will be the area of the triangle projedled on the 



ft 



plane * 2, which will be /' := - . In the fame manner, t 



2 



and /'' reprefenting the projeflions of the fame triangle on 

 the planes y 2, xy, we have 



t=\A,i-'= IC. 

 Problem III. 

 Given the co-ordinates of a point, and the equations 

 of a ftraight line, to find the equation of the plane which 

 pafl'es through the ftraight line and the given point, let 

 x',y', %' be the co-ordinates to the point ; 

 X =■ az -\- a '\ 



V =. bz + ^ > be the equations of the line. 



'b=^[x-a) = a[y-5)S 



Q Then 



