GEOMETRY. 



A {x ~ cc) + V, (y - r^) + a = o (e) 



A {x - «') + B (.V -$') + z = o (f) 



A and B being two conftant quantities determinable by the 

 following equations 



I ^ Aa + Bi = ol 

 I + Aa'+ Bb' = oS 



A - -T-. 7-r- U) 



hence 



B 



ab' - 

 a' - 



(2) 



ab' — a'b 



The perpendiculars to thefe parallel planes drawn through the 

 points P, P' have their equations 



I II . - . X— A-z + a.; y=B% + li 

 2d . . - .V = A 2 + k' ; ^' = B a + o' 

 The plane drawn through the firll of thefe perpendiculars, 

 and the firfl given line, has for its equation 



L (.V - «) + M (j - /3) + a = o (E) 

 L and M being given by the two equations 

 • I + L A + M B = o (3) 

 I+Lfl +Mi=:o(4) 

 The equation of the plane drav,-n through the fecond 

 perpendicular, and the fecond ftraight line is 



L' (.V - :•) -f M' {y _ ,9') + z = o {£') 

 L' and M' being determinable by thefe equatiooe 

 I + L' A + M B = o 

 I + V a' +W b' = o 

 Now each of thefe kill planes contains the required Ime, 

 therefore the equations of their line of interfcition will be 

 thofe required. 



The equations ( I ) (2) give the values of A and B, and 

 combining them with equations (3) (4) the following 

 values are obtained for L M, L' M' ; 



a - a' + b {ab' - a'b) 



L = 



L' = 



M = 



M'=: 



a [a' - a) + b' {b' - b) 

 a -a' + b' {ab' — a'b) 



a (a' - a) + b' {b' - b) 

 b - b' - a (ab' — a'b) 

 a {a' - 

 b - b' 



^) 



1- b{b' -b) 

 ' {af - ab) 



b' 



_: {a' -a) + b {b' - b) 

 Subflituting thefe values in equations (E) (E') we have 



(*■-«) ^a-a' + b {ab'-a'l)j + (y-i£)h 



- a (a b' - a' b)^ + z' |a {a' - a) + b {b' - b) I 



= 0; (.V- a') ^a-a' + b' {ab' - a' b)\ + 0'-/3) 



Sb - b' - a' {ab' - ab) | + z |a' («' - a) + b' 



(i' — ^ ) f =0; where the fecond equation may be dedu- 

 ced from the firfl by changing a,b,a,^ into a, b', a', /? , and 

 a', b' into a, b- 



Fionn thefe two equations which reprefent the required 

 line, its projeftions on the planes ^z, x z, may be found by 

 fiiccefrivcly ehminating x and ji. 



It remains now to determine the abfolute length of the 

 fhortcft diilance between the two lines. 



If from the origin of the co-ordinates a perpendicular be 



let fall on each of the parallel planes, thefe, having the fame 



direction, will become one and tlie fame ftraight line : their 



difference, or the diftance between the two planes, will be the 



fhortoft diftance required. The length of thefe perpendiculars 



will, by the laft problem, be 



A« + Bj3 



r = r r^ ^j for the plan« 



V U + A'- + B- ^ 



A (.» — a^ + B (j. — /S) -f z = o ; D being c= A cc 

 + B^; 



, „, A a' +■ B ,3' 



and P ^ ^ , tor tlie plane 



./ I + A- f B' '^ 



A {x — a') .f- B [y — p') + a — O, v.-herc D' = A a 



+ B;3'. 



1 he ditterence P — P = — j-; -frt 



,/ ^1 + A- + B ) 



fubftituting for A and B their values 



p, _ p ^ {-'--) jb' -b) - {^' -^) {-'-<- ) _ _, 

 ^/ { Cy - ay + {b' - by + {a'b - ab)j 



When the ftraight lines meet each other, this diftance being 

 nothing, {■!' — a.) {b' — b) - {■^' — ,3) {u' — a) = r; 

 the fame equation, as has been already found, expreffing 

 two right lines which interfeit each other. 



On the tra'isfonimlion of the Co-orilirintes — Given the co- 

 ordinates of a point in relation to three rectangular planes, 

 to determine the co-ordinates of this point, in rcfpect to 

 three other planes. 



Thefe three new planes being given in pofition in refpeft 

 to three primitive planes, their equations are given. 

 Let thefe be, for the 



ift, A .r -i- B J/ 4- C z + D = o 

 2d, A' X + W y +C' z -\r '^' = O 

 3d, A'' .V -i- B"^ + C" z -f- D" = o 



Thefe three planes interlcft each other two by two ire 

 three ftraight lines which are tlie new axes. The new co-or- 

 dinates of the point are meafured on the lines drawn through 

 this point parallel to the new axis. The length of any one of 

 thefe co-ordinates is the part of one of thele lines contained 

 between the point, and the plane of the co-ordinates to 

 which this line is parallel. 



Let .r,^,^, be the co-ordinates of the point in relation to the- 

 primitive planes, and u, v, -u.; its co-ordinates in relation to the 

 three new planes. I'or concifenefs let 



(A(CB'' - C'B') + B (A'C- A"C') 4- C 



(B'A"- B''A')» 



~ {C'B"-C"B'/-f(A'C"-A' C')M-(BA"-B"AV- 



(A'(C B" - C" B) 4- B' (A C - A" C) 4 C 



(B"A"- B"A )" 



" - ■(CB"-C'B)^4-(AC"-A"C)^4-(BA"-B''A>' 



A" (C B' - C B) 4- B" (A C - A' C)' + C" 



{BA'-B'A>' 



T m i ' 



" (CB'-C'B)^ 4 (AC-AC) 4 (BA'-B'A)» 

 The values of the new co-ordinates will be 



Ax+By+Cz-^B 



L" 



If the three new planes be fuppofed perpendicular to each 

 other, then A A' 4- B B' 4- C C = o' ; A A" -f B B" 

 4- C C" = o ; A' A" 4- B' B" 4- C C' = o. 



Multiplying the firft of thefe three equations by B", the fe- 

 cond by B', and fublrafting, we have C {C B" — C ' B) - 

 A (B'A" — B"A) = O. Multiplying the firft by A", 

 the fecond by A', and fubtradling, we have, 



B (B'A"- B'A') - C (A'C- A"C') = o. 

 Muhiplying the firft. by C", the fecond by C, and fub,- 

 traftiDff, we have 



^ A (A' 



