Royal Irish Academy. 533 



x and y to be used in the transformation of coordinates would be 



_x"(ax ' + by' - 1) 



x — - — , 



px' + qy' — 1 



y"(cx' + dy'-l) 



px 1 + qy' — 1 



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 

 a x + /3 y = 1 , 

 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 — /3. It appears from this, that if a and /3, 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 

 /3 be connected by an equation of the rath 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 ob- 

 vious manner as we enter upon the analytic geometry of the sphere. 



A spherical curve being represented by an equation between rec- 

 tangular coordinates, the equation of the great circle touching it at 

 the point x' , y' , is 



(y — y') dx' — (x — x') d y' = ; 

 the equation of the normal arc at the same point is 

 (y ~ V 1 ) [d y' + x' 0' dy' -y'd #')] 

 + (x - x') [dx' + y' (y'dx' - x' dy')~] = 0. 

 Now, if we differentiate this last equation with respect to x' and y' , 

 supposing x and y to be constant, we should find another equation, 

 which, taken along with that of the normal arc, would furnish the 

 values of x and y, the coordinates of the point in which two con- 

 secutive normal arcs intersect : and thus, as in plane geometry, we 

 find the evolute of a spherical curve. 



Let 2 y be the diametral arc of the circle of the sphere which 

 osculates a spherical curve at the point x\ y', Mr. Graves finds that 



tan 7 = ± ldx^ + dy^ + (x'dy'-y'dx')^ 

 ~ (1 + x h2 + y'~)i' (dx' d 2 y' — dy' d 2 x') 

 For the rectification and quadrature of a spherical curve given by 

 an equation between rectangular coordinates, the following formulae 

 arc to be employed : — 



d _ "/dx 7 ' 2 + dy' 12 + (*' dy' - y' d xj- 



1 S ~ 1 + x' 2 + y' 2 



y dx 



and d (area) = — 7 7 - a — -. 



v ' (1 + x 2 ) Vl + x 2 + tf 2 



In the preceding equations the radius of the sphere has been sup- 

 posed = 1 . 



