528 



CURVE LINES. 



Pure 

 CCXXVII. 



Fig. 14. 



on the other ; but if we now put the letters a, b, c, for 

 co-ordinates of the point in which the perpendicular 

 meets the plane, as the new values will be the halves 

 cf their former values, we have 



ax + iy-f C2=a'--f£ 2 -fc 2 

 for the equation of a plane. 



47. We may introduce into the equation the angles 

 which the perpendicular from the origin upon the 

 plane makes with the axes, instead of a, b, c. Let AM 

 (Fig. 14.) be the perpendicular, and M the point in 

 which it meets the plane; then MM'"=AP=a, MM"= 

 AQ=6, MM'=AR=c. Put « for the angle MAX, 

 /s for MAY, and y for MAZ; and put d for MA, the 

 distance of the plane from A. The angles MPA, MQA, 

 MRA, being right angles, we have AP=MAxCos. <* 

 AQ=MA x Cos. /3, ARrrMA X Cos. y, and AM 2 =AP 2 

 + PM*= AP* + PM' 2 + M'M 2 , that is in symbols 

 a—d Cos. «, b=d Cos. p,, c=d Cos. y, d 2 = eP + bj + c\ 

 Therefore, substituting in the equation, and dividing 

 by d, it becomes 



x Cos. x-\-y Cos. (S-\-z Cos. y =</. 

 By putting d Cos. <*, d Cos. /3, d Cos. y, instead of a, 

 b, c, in the equation a 9 -\-b' l -\-c % =d l , we shall find that 

 *, /3, y, are so related to one another, that 



COS. 2 tf-fCoS. 2 fi + COS. 1 y=l. 



48. Having given the equation of a plane, its position 

 may be determined from these formulas. Let the equa- 

 tion be Ax-f-B y + C2 + D=:0; by putting it in this 



ABC 

 form, rp-x + Y5J/ + TC J!: +l= > an d substituting A for 



and C for ^ , it will have this form, A ar-f- 



Curve 



Surfaces. 



D' Bf0r D' 



D' 



B i y-J-C2- r -l=0. Compare this with the expression 



Cos. a. Cos. /3 Cos. v * , 



— x — = y — -, 2 — r- +1=0, and we get 



d J d d T ' ° 



Cos. «.— — dA, Cos. /2= — dB, Cos. yz= — dC; from these 



equations we get d 2 ( A 2 + B 2 + C 2 )= Cos. 2 a, + Cos. 2 /3 4. 



Cos. 2 y= 1 , and rf=r — -— — — — •. As we now know 



a/(A 2 + B' + C 2 ) 



d, the distance of the plane from the origin of the co- 

 ordinates, and a, /3, y, the angles which a perpendicular 

 from the origin upon the plane makes with the axes, its 

 position is determined. 



The position of a plane may also be determined by 

 other data, particularly by its intersections with the co- 

 ordinate planes, which are called its traces; but our li- 

 mits do not admit of our enlarging on this subject. 



49- As the intersection of two planes is a straight line, 

 if those have given positions, the line will have a deter- 

 minate position. The position of a straight line in 

 space may therefore be expressed analytically by the 

 equations of any two planes which pass through the 

 line. Accordingly, the equations of a straight line re- 

 ferred to three co-ordinate planes are, 



A -r + B 7/4X2+1=0 .... (1.) 

 A'.r + B'# + C'2+1=0 .... (2.) 

 the variable co-ordinates x, y, z, being supposed the 

 iame in both. These serve to characterize the nature 

 of the line ; for by giving any particular value to one of 

 the co-ordinates, we can, by means of them, determine 

 the other two, and thence the point of the line corre- 

 sponding to the co-ordinates. 



The above equations, however, are not the only ones 

 by which the position of the line is determined, for we 

 may eliminate each of the three quantities x, y, z, in its 

 : urn. Let this be done, and, for the sake of brevity, let 



AB'— A'B=C1, CA'— C'A=B1, BC'_B'C=Al, 

 A — A' = A 2, B — W = B?, C — C = C2, 

 ;md we shall get 



Cly — Bl2+A2=0 .... (3.) 



Alz — Clx+B2=0 .... (4.) 



B1.t— Aly + C2=z0 .... (5.) 

 Any two of these equations (3.), (4.), (5.), may serve 

 instead of equations (1.), (2.), and they implicitly con- 

 tain the third. 



The equation (3.), which expresses the relation the 

 co-ordinates y, z, ought to have to one another for all 

 the points of the proposed straight line, belongs also to 

 a straight line traced on the plane YAZ, by letting fall 

 perpendiculars from every point in the line in question 

 on that plane, or to the intersection of the plane YAZ, 

 and a plane passing through the line perpendicular to 

 YAZ. The same is true also of equation (4.), in respect 

 of the plane XAZ, so that by chusing the system of these 

 two equations, the straight line proposed is considered as 

 the intersection of two planes respectively perpendicular 

 to the planes YAZ, XAZ. These planes, which are cal- 

 led the projecting planes of the straight line, because they 

 meet the co-ordinate planes to which they are perpen- 

 dicular in the projections of that line, are characterized 

 by equations (3.) and (4.), each considered by itself. 



In general, every equation containing two variable 

 quantities comprehended in the same co-ordinate plane, 

 ought to be regarded as belonging to a line traced 

 through the bottoms of an infinite number of perpen- 

 diculars erected on that plane. If all the perpendicu- 

 lars stand upon a single straight line, they will he in a 

 plane perpendicular to the co-ordinate plane. It must 

 also be remarked, that when only one of the co-ordinates 

 is determined, a plane is thereby indicated parallel to 

 that to which the ordinate is perpendicular. 



50. The equations of curve surfaces come next to be 

 considered after that of a plane. Let us take the sphere 

 as an example ; supposing the co-ordinates of any point 

 on its surface to be x, y, z, those of its centre to be a, 

 b, c, and its radius ~d, the expression we found in 

 article 45, for the distance between two points in space 

 gives us immediately 



(x—ay + (y—by + (z—cy=d*; 

 or, x t -\-y i +z' i + 2ax+2by-{-2cz=d i — a 2 — b l — c 2 , 

 for the equation of the surface of a sphere. 



If the centre of the sphere be at the origin of the co- 

 ordinates, then a, b, c, are each =0, and the equation 

 of the surface is simply x 2 +^ 2 -|-2 2 =:c? 2 . 



If a sphere be cut by a plane, it is known that the 

 section will be a circle ; now, as the co-ordinates of 

 every point in the circumference of this circle must sa- 

 tisfy at once the equations of the sphere and plane, the 

 position of a circle in space will be expressed by these 

 two equations, viz. 



Ax+By + Cz+D=0; 

 ( x —a) z +(y—by-{-(z—cY—d i =0. 



51. Let RKS (Fig. 15.) be a line of any kind traced p . 



on the plane X AY, the nature of which is known by ccxxVll 

 an equation between its co-ordinates AH and HK ; and pjg, 15. 

 let a straight line KM move in space, so as to be always 

 parallel to a line given by position, and at the same 

 time pass through the line RS ; by this motion, the line 

 KM will generate in space the surface of a solid which 

 may be called a cylinder, because of its analogy to the 

 cylinder of the elements of geometry. 



To find the equation of this kind of surface, let AP 

 r=x, PM"=y, and MM"=z, be the co-ordinates of any 

 point on it : Join KM", and draw KN parallel to AX. 

 Because the lines KM, MM" are parallel to lines given 

 by position, the plane KMM" is parallel to a plane gi- 

 ven by position, therefore KM", its intersection with the 

 plane XAY, is parallel to a line given by position; thus 

 the angles MKM", M"KN will each have a given mag- 



