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
... cosß , - - as y is the angle between Ox and Oy , and ẞ that between Ox and Oz . Similarly for the other coordinate axes we have - r cosμ = y1 − y + ( 2 , − 2 ) cosa + ( x , − x ) cosy , 1 r cos v = 21 - - 1 z + ( x , − x ) cosẞ + ...
... cosß , - - as y is the angle between Ox and Oy , and ẞ that between Ox and Oz . Similarly for the other coordinate axes we have - r cosμ = y1 − y + ( 2 , − 2 ) cosa + ( x , − x ) cosy , 1 r cos v = 21 - - 1 z + ( x , − x ) cosẞ + ...
Page 14
... cosẞ + 2 ( x , − x ) ( y1− y ) 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 ...
... cosẞ + 2 ( x , − x ) ( y1− y ) 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 ...
Page 15
... cosß , = A cosa , A = A cos ẞ , A = A cosy . Squaring and adding these , and observing that by the preceding theorem ... cosß , z = r cosy , x1 = r1 cosα1 , y1 = r , cosß1 , z1 = r , cosy1 , X1 = r , 1 therefore cose cosa cosa , + cosß ...
... cosß , = A cosa , A = A cos ẞ , A = A cosy . Squaring and adding these , and observing that by the preceding theorem ... cosß , z = r cosy , x1 = r1 cosα1 , y1 = r , cosß1 , z1 = r , cosy1 , X1 = r , 1 therefore cose cosa cosa , + cosß ...
Page 16
... cosẞ , r cosy . Then if a ,, B , Y ,, 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 , r cosß cosß , r cosy cosy , and their sum is r ( cosa cosa , + ...
... cosẞ , r cosy . Then if a ,, B , Y ,, 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 , r cosß cosß , r cosy cosy , and their sum is r ( cosa cosa , + ...
Page 239
... cosß , cosy , x ' - p2 = x = p cosa y ' - y = p cosẞ = 2 p * d2x ds2 21 day ds d2z - z = p cosy = p2 2 ds2 266. To calculate an expression for ᏧᎾ torsion . ds ' the measure of The equation to the osculating plane at the points x , y ...
... cosß , cosy , x ' - p2 = x = p cosa y ' - y = p cosẞ = 2 p * d2x ds2 21 day ds d2z - z = p cosy = p2 2 ds2 266. To calculate an expression for ᏧᎾ torsion . ds ' the measure of The equation to the osculating plane at the points x , y ...
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ax² axis centre chords coefficients condition cone consecutive constant coordinate planes cosines cosß cosv cosy curvature curve of contact cylinder determined developable surface dF dF dF dF dx diametral plane direction-cosines dx dy dy dz eliminate ellipse ellipsoid equal expression find the equation geometrical given line Hence hyperbolic hyperbolic paraboloid hyperboloid infinite number Let the equations line of intersection lines of curvature locus Multiplying normal plane origin osculating circle osculating plane P₁ parabola parameters perpendicular plane curve plane of yz planes parallel positive projection Px² quantities Qy² r₁ radii radius ratios rectangular ruled surfaces Rz² second degree second order sections shew singular points sphere straight line substitute suppose surfaces of revolution tangent plane three equations values vanish variables x₁ y₁ y²² zero