1 80 Mr. T. S. Davies on Bernoulli's Solution 



vantage over the other method. Had, therefore, the mere so- 

 lution of the problem been the object to be principally sought 

 for, there can be no doubt that it is already rendered as simple 

 as it ever can be; and a very ample list of such solutions may 

 be consulted in Professor Leybourn's edition of that valuable 

 but unpretending little periodical, the Ladies' Diary, vol. iv. 

 p. 314. We find there the solutions of Messrs. Ivory, Wallace, 

 and Skene, as well as the very amplified discussions of De- ' 

 lambre, as given both in his Astronomie, and in his Histoire 

 de V Astronomie Ancienne. Two other solutions by Hachette, 

 printed in the Correspondence of the Polytechnic School, 

 vol. ii. ; and others still, by Keill, Lemonnier, Maupertuis, 

 Emerson, Mauduit, Cagnoli, &c, may be found in the respec- 

 tive writings of those authors. All these, however, are of the 

 kind which may be called geometrical, in contradistinction to 

 the method of Bernoulli and D'Alembert. 



Of these, Mr. Skene's is the only one which takes the pro- 

 blem in the more general form, in which another almacantar 

 is substituted for the horizon of the common problem ; and 

 his solution is truly elegant as a specimen of the method of re- 

 search which he adopted. Unfortunately it confined him to 

 the simple question of the minimum, and he also was thereby 

 led to give an incomplete solution, compared with that fur- 

 nished by the calculus. The two cases, the maximum and the 

 minimum, are so intimately connected that they present them- 

 selves naturally as the extreme cases of the question, When the 

 duration of twilight is given, tojind the declination^ and they 

 appear together in the same final equation interwoven in a 

 single formula, and demanding in all respects equal attention. 

 On this account, elegant as the solution of Mr. Skene is, it 

 does not remove the necessity for reconsidering the problem 

 under another aspect. 



Some time ago, when composing a paper (since published 

 in vol. xii. of the Edinburgh Transactions,) on the application 

 of great-circle co-ordinates to the investigation of spherical 

 loci, I was led to examine a considerable number of those 

 problems which had been subjects of discussion, by the older 

 methods of treating spherical curves — and the problem of 

 shortest twilight amongst the rest. As, however, that pro- 

 blem did not properly appertain to my plan, I relinquished the 

 inquiry for others more interesting to me at that moment, and 

 laid aside the few memoranda which I had made concerning it. 

 A day or two ago my attention was accidentally recalled to 

 the subject ; and as in those memoranda the problem had been 

 viewed under its most general aspect, and the true character 

 of all the roots of the equation assigned, — as, moreover, the 



