416 Mr DAVIES on the Equations of Loci 



rather looks upon them as two distinct curves, whose ultimate tangents make 

 an angle of 180 with each other. Perhaps my mode of considering it might 

 justify such an objection that they were two distinct curves, whose gene- 

 rating azimuths were the supplements of each other, and proceeding con- 

 trary ways from the meridian of reference ; and, therefore, it becomes de- 

 sirable to examine the question a little more closely. This I propose to do 

 in the present note. 



The differential equations in XXXI., viz. d 6 cot a = d (p cosec might 

 be written 



d(b dd> 



+-cot a d6 = -*--- = . /. ,. 

 ~ sm0 sin (-T- <p) 



and its integral will therefore be 



log tan f $^\ = cot a . 6 + const (3') 



or since, as is found in the passage quoted, the const 0, and 



log tan (it -^-) = cot a (V) 



This expresses two equal and symmetrical branches of the same curve re- 

 ferred to the same polar co-ordinates, in short, the points which geogra- 

 phers denominate the Periceci ; and being included in the same equa- 

 tion, constitute mathematically continuous branches of one curve, and not 

 merely tangential branches of two separate curves. The validity of the in- 

 ference is hence fully established, for these two numbers are those which, 

 in my former paper, were shewn to meet in the pole ; and hence the ship 

 would return to the same point after pursuing such a course as was there laid 

 down. 



I was desirous of inserting here some further properties of the loxodrome, 

 especially the analogues to JAMES BERNOULLI'S properties of the logarith- 

 mic spiral * : but the space which I can allow myself does not allow me to 

 do so, especially as I wish to notice here a new curve, having certain remark- 



* Opp. torn ii. p. 491. 



