upon the Antique Dials. 99 



tact became 90. We, however, shall have occasion to speak more amply 

 on the subject farther on, and therefore dismiss it for the present. 



If, on the other hand, we had used if, we should have had, when x = 0, 



if = tan I cosnL, 



the equation of the hectemorial equatorial cylinder (X1I1.), into which cy- 

 linder the cone is now transformed. 



But if *. = 90, then v' attains an infinite value, which shows that the 

 intersection of the cone (that is, in this case, of the polar tangent plane) 

 with the equator is become infinitely distant. 



XVII. 



When we develope a cone, the radius of the sector into which it is 

 evolved is the arete of the cone itself. The radius of that sector is to the 

 radius of the circle of contact (as HF : FO : : HG : GK : :) as cosec A : 1. 

 Hence all the values of L, reckoned from the beginning of the longitude, 

 are diminished in the sector in the same ratio. Hence, to express the equa- 

 tion of the hectemoria upon such a conic surface when developed, we must 

 take this change into account. Hence, if 8 be the angle found between the 

 radius sector v, and the origin of angular co-ordinates, the equations for v 

 and if will become, 



a cosec 2 X cot I 



~ cot X cot I -J- cos (n . cosec X . 9) 



. a cosec X sec X cos (n . cosec ^ . P) 



cot X cot I + cos (n . cosec * . 0) 



The latter, v', being always reckoned from the developed equator towards 

 the centre of the sector. 



XVIII. 



These equations may be reduced wholly into factors, and thereby 

 adapted to logarithms, though it will upon experiment prove of little ad- 

 vantage to do so, if facility of calculation be our object In construction, 



