14<2 Royal Society of Edinburgh. 



selves. In February 1S30, Mr. Davies transmitted to the Royal So- 

 ciety of Edinburgh a paper containing a complete investigation of 

 the course of these lines upon the sphere, as derived from a spherical 

 equation of the curves themselves, between the variable right ascen- 

 sion and declination of the points which compose it. Of the various 

 deductions and methods contained in that paper we do not propose 

 here to give any account, as the paper itself forms part of the volume 

 which the Society will shortly publish : and we allude to it merely as 

 furnishing the motive which induced the author to examine the ge- 

 neral character of spherical loci by means of equations between the 

 spherical co-ordinates of their points. 



The first part of the paper is devoted to establishing the equations 

 of great and less circles, so as to fulfill certain assigned conditions, — 

 such as passing through given points, making given angles with given 

 great circles, the transformation of the equations which result from 

 change of origin and direction of the axes of co-ordinates, &c. Amongst 

 these we noticed a curious theorem, tvhich assigns the area of a sphe- 

 rical triangle in terins of the co-ordinates of its three angles. The chief 

 part of the paper was, however, devoted to other loci, and the results 

 which the author obtained were well calculated to show the value of 

 the method which he employed. We can only allude to a few. The 

 locus of which the oval window of Viviani, the spiral of Pappus, and 

 the sun's annual vertical path over the earth, were cases dependent 

 upon the values of the constants, was discussed at great length ; as 

 was the spherical epicycloid, in which it was shown that the case 

 pointed out by John Bernoulli was the only one whose arcs were rec- 

 tifiable. The loxodrome was also investigated at considerable length, 

 and amongst the results obtained was an affirmative answer to the 

 q\iestion, " Whether a ship sailing on a particular rhomb would ever 

 return to the same place ?" — a question which in former times was 

 often unsatisfactorily discussed. Mr. Davies showed that it would 

 return after passing through both poles. The beautiful theorem of 

 Lenell, which states that " the locus of the vertex of a spherical tri- 

 angle whose base and area are given is a circle," was here amongst 

 others, for the first time, the author thinks, investigated in a legiti- 

 mate manner : this consisted in finding the equation of the vertex, and 

 showing that this equation represented a circle, — a process directly 

 opposed to those hitherto employed, they being calculated only to de- 

 termine whether a previously suspected locus be the true one, not to 

 discover from first principles what the locus actually is. 



Mr. Davies also touched upon the spherical conic sections, though 

 but slightly, on account of his intention of treating the subject at 

 length in a separate dissertation. The extent to which the present 

 paper had already gone prevented his doing more than mentioning 

 one or two of their properties ; especially as the complete investiga- 

 tion required him first to lay down some theorems respecting the dif- 

 ferental co-efficients which enter into the expression of the tangents, 

 normals, Jcc. : and as this would still further have lengthened the 

 paper, he judged it better to defer them till a future occasion. 



The paper has been ordered by the Council of the Society to be 

 printed in the forthcoming (twelftli) volume of its Transactions. 



BELFAST 



