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 13
... cosy + ( 2 , −z ) cosß , 1 - 1 asy is the angle between Ox and Oy , and ẞ that between Ox and Oz . Similarly for the other coordinate axes we have r cosμ = y1y + ( 2 , 2 ) cosa + ( x , x ) cosy , r cos y = 21 - - - z + ( x , − x ) ...
... cosy + ( 2 , −z ) cosß , 1 - 1 asy is the angle between Ox and Oy , and ẞ that between Ox and Oz . Similarly for the other coordinate axes we have r cosμ = y1y + ( 2 , 2 ) cosa + ( x , x ) cosy , r cos y = 21 - - - z + ( x , − x ) ...
Page 14
... cosy . It is obvious that this gives us the expression for the length of a diagonal of a parallelepiped in terms of the sides and the angles which they make with each other . 16. To find the relation between the cosines of the angles ...
... cosy . It is obvious that this gives us the expression for the length of a diagonal of a parallelepiped in terms of the sides and the angles which they make with each other . 16. To find the relation between the cosines of the angles ...
Page 15
... cosy . x Squaring and adding these , and observing that by the preceding theorem cos'a + cos B + cos y = 1 , 2 we ... cosy , z1 = r , cosy ,, 21 therefore cose cosa cosa , + cosẞ cosß , + cosy cosy ,. 19. This theorem proves the ...
... cosy . x Squaring and adding these , and observing that by the preceding theorem cos'a + cos B + cos y = 1 , 2 we ... cosy , z1 = r , cosy ,, 21 therefore cose cosa cosa , + cosẞ cosß , + cosy cosy ,. 19. This theorem proves the ...
Page 16
... cosy . Then if a B1 , Y1 , be the angles which the second line makes with the axes , the projections of the preceding quantities on the second line are r cosa cosa , and their sum is r cosß cosß ,, r cosy cosy , r ( cosa cosa , + cosß ...
... cosy . Then if a B1 , Y1 , be the angles which the second line makes with the axes , the projections of the preceding quantities on the second line are r cosa cosa , and their sum is r cosß cosß ,, r cosy cosy , r ( cosa cosa , + cosß ...
Page 17
... ABC makes with the planes of yz , zx , and xy ; hence , by the last article , if a , ß , y , be these angles , A A x COSC = A ' cos B = 41 , A Α . cosy = A A · C But if we project the broken line ONMA on OH FUNDAMENTAL THEOREMS . 17.
... ABC makes with the planes of yz , zx , and xy ; hence , by the last article , if a , ß , y , be these angles , A A x COSC = A ' cos B = 41 , A Α . cosy = A A · C But if we project the broken line ONMA on OH FUNDAMENTAL THEOREMS . 17.
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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...