X =. 



128 



x"{ax' -\-by'-l) 

 px + qy'—l 



_ y"{cx' + dy'-\) 

 ^ ~ px' + qy'—l 



In which a, b, c, d, p, and q, are functions of m, m', x", 

 and y". It is evident that the degree of the transformed 

 equation in x', y', will be the same as that of the original 

 one in x and y. 



The great circle represented by the equation 



ax -\- ^y = \, 



meets the arcs of reference in two points, the cotangents of 

 whose distances from the origin are a and /3 ; and, if the 

 arcs of reference meet at right angles, the coordinates of the 

 pole of this great circle are — a, and — j3. It appears from 

 this, that if a and j3, instead of being fixed, are connected by 

 an equation of the first degree, the great circle will turn round 

 a fixed point. And, in general, if a and j3 be connected by 

 an equation of the w*'* degree, the great circle will envelope 

 a spherical curve to which n tangent arcs may be drawn from 

 the same point. Thus, the fundamental principles of the 

 theory of polar reciprocals present themselves to us in the 

 most obvious manner as we enter vipon the analytic geometry 

 of the sphere. 



A spherical curve being represented by an equation be- 

 tween rectangular coordinates, the equation of the great 

 circle touching it at the point x', y', is 



iy — y') dx' — (x — x') dy' = ; 



the equation of the normal arc at the same point is 



(2/ - .VO W + x' {x'dy' - y'dx')] 

 + (x— x') [dx' + y' {y'dx' — x'dy')] = 0. 



Now, if we diflferentiate this last equation with respect to 



