286 Mr DAVIES on the Equations of Loci 



by (8, IV.) And carrying cot a, cot\ into the term within the vinculum 

 in (10, IX.), we shall have the same form as that given in (9, IV.) 



Cor. 3. If , = \, not taking ft = *,, we shall have (10) converted 

 into 



cot 2 a, tan a, 



cot (p = {cos 6 (sin ft sin /c y ) sin 6 (cos ft cos K.) } 



sin ft K, 



2 cot a, sin ^' K J. 



__ 



sin ft -K, 



................. ' 



2 



which is the equation of a circle perpendicular to the given one, and such 

 that the centres have the same latitude. 



Cor. 4. If we seek that one of this class of circles which belongs to 

 the limit of (3, K,, we shall find it by putting ft = K, in equation (12). 

 This will give 



cos ~~ =.!> and cos(^^ _ 6\ = cos (ft 6), 



and the equation is, 



cot (f) = rfc cot a, cos (ft 6) ................ (13) 



Cor. 5. If we had taken originally ft = K, , establishing no other rela- 

 tion amongst the constants, for the present, we should have had 



:ot (f) = qp - {cos 6 sin ft (tan a, tan XJ sin 6 cos ft (tan a, tan \) } 



Sill \J 



_ _ cot \ cot a, sin (ft 6) (tan q ; tan \) 



sin 0" 



(14 



This equation gives rise to the following remarks : 

 Imo, The denominator being essentially zero, if the four factors in the 

 numerators be finite, the value of cot $ is necessarily infinite, and hence 



