A treatise on the application of analysis to solid geometry, commenced by D.F. Gregory, concluded by W. WaltonJ. Deighton, 1852 - 310 pages |
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Page 22
... constants in the equations ( 3 ) ; for , considering the left - hand equation , we see that it is the equation to a straight line in the plane of xz , which is evidently the projection of the given line on 22 THE STRAIGHT LINE .
... constants in the equations ( 3 ) ; for , considering the left - hand equation , we see that it is the equation to a straight line in the plane of xz , which is evidently the projection of the given line on 22 THE STRAIGHT LINE .
Page 23
Duncan Farquharson Gregory. which is evidently the projection of the given line on that plane . Now a is the tangent of the angle at which this projection cuts the axis of z , and p is the portion of the axis of x intercepted between the ...
Duncan Farquharson Gregory. which is evidently the projection of the given line on that plane . Now a is the tangent of the angle at which this projection cuts the axis of z , and p is the portion of the axis of x intercepted between the ...
Page 41
... evidently equal to the difference of the perpendiculars from the origin on the planes ( 3 ) and ( 4 ) . That difference is - ± { ( a − a ' ) λ + ( B − B ' ) μ + ( y — y ' ) v } ; so that , if ♪ be the required distance , = ± - ( mn ...
... evidently equal to the difference of the perpendiculars from the origin on the planes ( 3 ) and ( 4 ) . That difference is - ± { ( a − a ' ) λ + ( B − B ' ) μ + ( y — y ' ) v } ; so that , if ♪ be the required distance , = ± - ( mn ...
Page 42
... evidently the same as those found in Art . ( 50 ) , and we have only to determine x ' , y ' , z ' : now these being the coordinates of an arbitrary point in the line , we may assume that point to be the intersection of ( 1 ) and ( 3 ) ...
... evidently the same as those found in Art . ( 50 ) , and we have only to determine x ' , y ' , z ' : now these being the coordinates of an arbitrary point in the line , we may assume that point to be the intersection of ( 1 ) and ( 3 ) ...
Page 45
... evidently l ' cosλ + m ' cosμ + n ' cosy = 0 ; or ll ' + mm ' + nn ' + a ( mn ' + m'n ) + b ( nl ' + n'l ) + c ( lm ' + l'm ) = 0 ; and the conditions that they may be parallel are 1 + bn + cm m + an + cl n + am + bl l ' + bn ' + cm ' m ...
... evidently l ' cosλ + m ' cosμ + n ' cosy = 0 ; or ll ' + mm ' + nn ' + a ( mn ' + m'n ) + b ( nl ' + n'l ) + c ( lm ' + l'm ) = 0 ; and the conditions that they may be parallel are 1 + bn + cm m + an + cl n + am + bl l ' + bn ' + cm ' m ...
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Common terms and phrases
axis centre chords coefficients condition cone constant coordinate planes cosines cosn cosß cosv cosy curvature curve of contact cylinder determined developable surface dF dF dF dx diametral plane direction-cosines dv du dv dx dy dz dx² eliminating ellipse ellipsoid equa equal expression find the equation formulæ geometrical given line Hence homogeneous function hyperbolic hyperbolic paraboloid hyperboloid infinite number Let the equations line of intersection lines of curvature locus Multiplying normal plane origin osculating circle osculating plane parameters perpendicular plane curve plane of yz planes parallel positive projection Px² quantities Qy² ratios rectangular ruled surfaces Rz² second degree second order sections shew singular points sphere straight line substitute tangent plane three equations values vanish variables x₁ x²² Y₁ y²² zero
Popular passages
Page 16 - To express the area of a triangle in terms of the coordinates of its angular points.
Page 307 - A Treatise on the Application of Analysis to Solid Geometry. Commenced by DF GREGORY, MA, late Fellow and Assistant Tutor of Trinity College, Cambridge ; Concluded by W. WALTON, MA, Trinity College, Cambridge.
Page 10 - It, so that PR is equal to MN. Now the inclination of a straight line to a plane is the angle which the line makes with the intersection of the plane and a plane perpendicular to it passing through the line. Since, then, PM and QN are perpendicular to ABCD, the plane of PQMN is also perpendicular to it, and the inclination of PQ to the plane AB CD is measured by the angle between PQ and MN or the equal angle QPR.
Page 280 - The sum of the squares of the projections of any three conjugate diameters on a fixed line is constant. Instead of projecting the diameters on the line directly, it is better to project the coordinates of the extremities of each diameter, and add them. Now, if X...