VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 473 



altit. is very small, and there is also a considerable uncertainty with respect to the 

 latit. this error will probably become of more importance, and we may find it 

 necessary to guard against it. Now it is manifest that, by diminishing the multi- 

 ple which the area gb is of the area gc, exactly in the same proportion we shall di- 

 minish this error. 



Mr. L. further adverts to the methods of correcting or avoiding certain small in- 

 accuracies, which however may commonly be omitted, or obviated ; he then pro- 

 ceeds : I have hitherto supposed that this method is only to be adopted, when the 

 sun, at each observation, is within 1 5° of the meridian ; or, to speak, more accu- 

 rately, when both the azimuth and the hour-angle are so small, that we may con- 

 sider their tangents as bearing a given ratio to each other ; and indeed these limits 

 should never be transgressed, when it can possibly be avoided ; for we have seen 

 that, whatever be the method employed, the smaller the hour^angle, the greater is 

 the exactness with which the lat. is determined. Sometimes however it will be im- 

 possible to make both, or perhaps either of our observations within the distance re- 

 commended ; but even in these cases our rule may be conveniently applied. It has 

 already been demonstrated that we can never be subject to any material error in 

 consequence of the inequality of the areas gb and gb, except when the zenith-dis- 

 tance of the sun, at his meridian altitude, is very small ; and for this case an 

 effectual remedy has been provided. 



Before concluding the theory, it may be observed, that though I have directed 

 the altitudes to be taken on different sides of the meridian, it is by no means re- 

 quisite that we should invariably adhere to this precept. We have seen the reason 

 indeed why it is expedient, in most instances to prefer this method, being generally 

 calculated to produce a much greater degree of exactness in the result. This how- 

 ever is not always the case ; for, if one of the observations be made beyond the dis- 

 tance originally prescribed, it is of little importance whether the 2d altitude be 

 taken on the same side of the meridian, or not. But it will sometimes be im- 

 possible to make the observations on different sides of noon ; and hence it becomes 

 necessary to inquire in what manner the real latitude may be discovered in these cir- 

 cumstances. The clock gives us the interval between the observations equal to 

 ae — ae ; and by computation we find ag and ag, and thence we deduce eg — eg, 

 the difference between the errors in time. Having then assumed, without any re- 

 gard to accuracy, 1 portions of time, corresponding to the 1 observations, whose 

 difference is the same as the difference between the errors before determined, and 

 which are to each other in the inverse ratio of the hour-angles, we must increase or 

 diminish them both equally, till we get the areas in the first table of the same mag- 

 nitude, and then we may conclude that we have obtained the proper value of each. 

 The directions which have been given for the prevention of errors in the former 

 case, when the altitudes are taken on different sides of the meridian, are very easily 

 accommodated to the present ; and it would therefore be superfluous to bestow any 

 further consideration on them. 



VOL. XVIII. 3 P 



