342 Mr DA VIES on the Equations of Loci 



We shall take a few particular cases for illustration. 



}) yiii/i bi ii;i . :-. ol 



Imo, Let = ; that is, let the projection become the plane of the 



equator. 



Then (a). Let the projecting point M coincide with c, or the projection 

 be gnomonic ; we shall have, in this case, 



1 a ke _ a u \ 



-= -- , or - = H- inf. or v = .......... (3.) 



v 2axo v 



agreeing with what we know ought to take place, as the only intersection 

 of the projecting cone with the plane of projection, is now the centre itself, 

 indicated by v = 0, for all values of 0. 



(b). Let M coincide with B ; then b ""a, and we have 



which is the equation of the logarithmic spiral, agreeing with the well 

 known result of Dr HALLEY *. and of JAMES BERNOULLI f. 



(c). Let M be at A ; then 6 - a, and the formula becomes 



1 2 alf? 



~ rV = aJ <- 1 



The reversed branch of the logarithmic spiral: 

 (d). Let b = infinity. Then 



(6.) 



which is the equation of the orthographic projection of the rhumb-line on 

 the plane of the equator. It is independent of the value of c, that is, the 

 projection is the same through whatever point of the axis the parallel to the 

 equator is drawn, as it ought to be. 



Phil. Trans. 1696. Abt. vol. i. p. 577., or New Abt. vol. iv. p. 68. 

 t JAC. BERN. Op. Omn. Ed. Cramer, torn. ii. p. 491. 



