154 U. S. COAST AND GEODETIC SURVEY. 



The deformation produced around each point is subjected 

 to a law which depends neither upon the nature of the 

 surfaces nor upon the method of projection. 



Every representation of one surface upon another can 

 be replaced by an infinity of orthogonal projections each 

 made upon a suitable scale. 



We note, first, that there always exists at every point 

 of the first surface two tangents perpendicular to each 

 other, such that the directions which correspond to them 

 upon the. second surface also intersect at right angles. 

 In figure 42 let CE and OD be two tangents perpendicular 

 to each other at the point on the firet surface; let C'E^ 

 and O'D' be the corresponding tangents to the second. 



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Fig. 41.— a curve and its projection. 



Let us suppose that of two angles C'O'B' and B'O'E' the 

 first is acute, and let us imagine that a right angle having 

 its vertex at turns from left to right around this point 

 in the plane GBE^ starting from the position GOD and 

 arriving at the position DOE, The corresponding angle 

 in the plane tangent at to the second surface will first 

 coincide with C'O'D' and will be acute; in its final position 

 it will coincide with D'O'E' ^ and will be obtuse; within the 

 interval it will have passed through a right angle. There- 

 fore, there exists a system of two tangents satisfying the 

 condition stated, except at certain singular points. From 

 this property we conclude that in every system of repre- 

 sentation there is upon the first of the two surfaces a 

 system of two series of orthogonal curves whose projec- 

 tions upon the second surface are also orthogonal. The 



