34 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



%i, %2, x 3 , #4 denote the distances of a point P from the four sides of a tet- 

 rahedron (the tetrahedron of reference); /i, wi, n\\ h, wh, n%', 1 3 , Ws, n$\ and 

 / 4 , w 4 , n* the direction cosines of the normals to the planes x\ = 0,0:2 = o, x 3 = o, 

 #4 = with respect to a rectangular system of coordinates x, y, z; and d\, d 2 , d 3 , 

 di the distances of these 4 planes from the origin of coordinates : 



xi = hx + m\y + iz di 



X 2 = /2# 



x 3 = I 3 x 



1, ^2, ^3, and 5 4 are the areas of the 4 faces of the tetrahedron of reference 

 and V its volume : 



37 = XiSi + XzS 2 + #3^3 + ^4^4. 



By means of the first 3 equations of (i) x, y, z are determined : 



x = AIXI + BiXz + CiX B + Di, 

 y = A 2 xi + B 2 xz + C 2 #3 + A, 

 z = A&i + ^3^2 + C 3 o;3 + A. 



The equation of any surface, 



F(*,;y,z) = o, 



may be written in the homogeneous form : 



F\ \AiXi + J5i#2 + Ci# 3 + p (5i^i + 5 2 ^2 + ^3^3 + ^4^4) , 

 -4 2 *i + ^2^2 + C 2 ^3 + p (^lOPi + S 2 # 2 + ^3^3 + ^4^4) , 

 I ^3^1 + ^3^2 + C 3 .T 3 + p (5ii + 5 2 rV 2 + 5 3 ^3 + 5 4 ^ 4 ) | = O. 



f PLANE GEOMETRY 



2.100 The equation of a line: 



Ax + By + C = o. 



2.101 If p is the perpendicular from the origin upon the line, and a and /3 the 

 angles p makes with the x- and y-axes : 



p = x cos a + y cos j3. 



2.102 If a! and /3 r are the angles the line makes with the x- and ;y-axes : 



p = y cos a! x cos /3'. 



2.103 The equation of a line may be written 



y = ax + b. 



a = tangent of angle the line makes with the #-axis, 

 b = intercept of the y-axis by the line. 



