GEOMETRY. 



*/x*+ y" + b", the other, as found in the preceding 



. a X-' -\- b y' + z' 

 problem, is — ' — ~ . 



^^ 1 + a- + y- 



Therefore the cofine of the required angle is equal 

 ax' + by' + x' 



But x' =■ a' a', y = i' x' ; therefore the cofine of the 

 angle formed by the two given ilraight lines 

 I + aa' + 6 b' 



co-ordinate planes, then * = i A, /' = i B, /'' = * C ; — 



6 

 being the fohdity of a pyramid which has for its bafc the 

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

 Now the folidity of this pyramid is the produd of the bafe 



T, by one third of its height ^ ; there- 



fore V = T 

 6 



D 



VA'- + B" + C' 



or fubfUtuting for 



^^ I + a'- + i' X V 1 + a" + b'~ 

 It appears from this expreffion, that when two ftraight 

 lines have for their equations 



f I X =z a z, y z=i b T. 



\2 X = a' z, ^ = i' z 



if they are perpendicular to each other, the following equa- 

 tion of condition will be obtained, i f oa' -f i A' = o, which 

 equation mav be likewife obtained direftly as follows : the 

 plane perpendicular to the firft line drawn through the origin 

 of the co-ordinates has for its equation ax + by-rz=^0. 

 But the perpendicular to the firft ftraight line muft be con- 

 tained in the plane perpendicular to it ; therefore the equa- 

 tions to the perpendicular x =. a' z', y = &' z, and the 

 equation of the plane muil fubfift at the fame time ; there- 

 fore I -t- a a' + i i' = o. The angle of the two planes may 

 be thus determined. Let ax i- b y + :ii — C, a' x -\- b' y 



3 ■/ A^ 4- B' + C 

 A, B, C their values 2 /, 2 /', 2 /", T' = /" + / -|- /'". 



If S be the area of another triangle whofe projections arc 

 /, t', s"j and fituated in the fame plane as the triangle T ; theo 

 S' = S' -f S" -f- S'« 



I 



SinceT = i ^/A^-t. B'+C'-r=(-~= 



' \ ^^ A- 



like 



T 

 manner — = 



+ B= -l-C 







(- 



B 



A-- + B-- 



s) 



T 

 t" 



%/ A^ + B-- + C 







which fignifies that any triangle it 



._-___ . to its projeftion on one of the co-ordinate planes, as radius 



-f s = C be the equations of the "planes ; thefe make with '' "^^ the cofine of the angle which the plane of the triangU 



each other the fame angle as the ftraight lines which are per- 

 pendicular to them, and which are drawn from the origin of 

 the co-ordinates; therefore the cofine of the angle formed 

 by the two given planes, is 



bb' 



1 + aa' + 

 + b X V 



makes with the plane on which it is projefted. 



But the triangle S being in the fame plane with the triangle T 



— ~±—-£. ^ _ '" 



T~'S'T~S' T-T*' 



therefore if the equation T' = /' -f- /" -|- ('" be put 



i' 



I + a'" + i' 



If the angle be required between one ftraight line and one 

 plane, then iuppofe a parallel drawn to the given lines 

 through the origin of the ordinates, and a perpendicular to 

 the plane, the angle contained between tlicle two ftraight 

 lines will bethe complement of the required angle ; and conle- 

 quently, the cofine of the angle of the two lines is the fine of angle R, whofe projeftions on the reftangular planes 



under this farm, T= = /+ =/'-)--— /", it will become 

 T S = ^ J -I- i',' A- t''s"; but (T' -f S)' = T' -f- 2 T S -t- 



S' = /' + /'" -t- t'"- + Zts + 2 t'l' + 2 t"s" + J^ + 



/' +s"'i therefore (T + S)' = (/ -f sY + (<' -|- t^ + 

 Taking in the fame manner in the fame plane a third tri» 



the angle required. 



The ftraight line, whofe equations are x = a z, y =: ix, 

 makes with the axes x, y, e, angles whofe cofines are 



V x^ -f 



y- + % 

 a 



V . 



+ y' 

 b 



+ s' 



-I- / + ^^ 



I 



I + a 



+ h' 



■^ I ■\- a'- + b' ^'' I -I- a' f i" 

 The fame expreifions are the values of the cofines of the 

 angles which a plane perpendicular to the ftraight line, and 

 whole equation is a v -r- i_)' -f s = o, makes with the co- 

 ordinate planes xy, zy, x a. If the equ;\tioii of the plane 

 uAx+B_)' + Cz-|-D=C, th'' cofines of the angles 

 which it makes with the co-ordinate p\anes are 



A B C 



■/A'- + B -t- C= v' A^ + B' + C^' ^' A; + B + C 

 and the expreffion found above for the perpendicular, let fall 

 from the origin of the co-ordinates on the plane, becomes 

 D 



V A- + B' + C 



It has been already remarked, that if T be the tri- 

 angle formed by the three lines which join, two and two, 

 (be three given poiatt, and ;, t', f it« ptojedtione on 



the 



r, r, r', it may be (hewn that (R + S -f T)' = 

 [r + s -[- t)l + (r' + j' + t'Y + (r" + /' + i'T i hence 

 if any plane figure whatever be projefted on three reftangu- 

 lar planes, the fquare of the area of this figure will be equal 

 to the fum of the fquares ot the areas of its three projedlioni. 



Problem V. 



Two ftraight lines being given, ift, to determine the 

 equations to a ftraight line perpendicular to each of them 

 on which their fliorteft diftance is meafured ; 2d, to fini 

 a:n exprefTion for this (horteft diftance. 



The direction of a plane parallel to two ftraio-ht line* 

 given in pofition may be determined: this plane being drawn 

 through any point in fpace, we may conceive a plane to 

 pafs tlu-ough each of the ftraight lines perpendicular to it ; 

 the interfeftion of thefe two planes is evidently the line 

 required, therefore the equations to thefe planes will be 

 thofe of the fine required. 



I.ct .1- = az -)- a, ji = i K -f 5, be the equations to the 

 firft line, it will meet the plane xjr in a point P, of which 

 the co-ordinates a =:: o, _y := /S, .r := a. 



The fccond ftraight line having for its equations .v ■= a' % 

 4- a', ^' = i' z 4- (5, it meets the plane xy iu a point P', 

 wliofe co-ordinates are c =r o, _y =: o, .v = at. 



The equations of the planes drawn through the points P 

 and P' parallel to the two given ftraight lines are of the form 



S^' 



A (x - 



