278 



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



sun would cause the time of passing from one given almacan- 

 tar to another to be a maximum or a minimum ? The actual 

 path of the sun is no part of the data; and hence D'Alem- 

 bert's difficulty here arose from tacitly introducing a new con- 

 dition into the solution of the, problem, which had not been 

 introduced into the process for finding the equation of the 

 problem. It is true that under the physical circumstances 

 which were known to exist (but which were unknown in the 

 problem in the aspect under which he viewed it as a mathe- 

 matician,) there was an obstacle to the maximum answer taking- 

 place actually in certain latitudes ; but in other given latitudes 

 the maximum might, with the present position of the ecliptic, 

 be actually attained. His showing that the maximum could 

 not be attained in certain particular cases, did not show that 

 it was unattainable in all : and hence his inference, as a ge- 

 neral one, that the second answer did not refer to the maxi- 

 mum case of the mathematical problem which he had solved, 

 was in all respects inaccurate. It is important to keep in 

 mind the fact, that the mathematical problem which we have 

 solved is not always identical with the physical problem which 

 we had proposed. 



Delambre has given as a re- 

 markable property of the azi- 

 muths at the beginning and 

 end of twilight, in the case of 

 the minimum, that they are sup- 

 plementary ; but this is not con- 

 fined to the case in which the 

 horizon is one of the almacan- 

 tars, as the following investiga- 

 tions show : 



1 . Let us take p t + p lr Then 



T^r/^T cos p — cos A cos p. 

 cos PZN = h — t-t-s *-/ 



sin A sin pi 



bs cot A a + cos^cos^— sin p # sin P ll ■ 



cos' 



as cot A 



= COt A 



sm p / (cos p t + cos pii) 



1— s in ftsin p u — (1— sin 2 pj) 



sin^ (cospZ+cosft,) 

 sin pi — sin p u 

 cos pi -f cos pu 



= cot A tan \p—p 4 

 And in the same way we find 



(14) 



