162 Astronomical and Nautical Collections. 



since all of therp neglect some small quantities which are hot 

 absolutely evanescent ; secondly, that they are all, though in 

 different degrees, much more troublesome than could be desired 

 for a preliminary calculation ; and, thirdly, that since no equa- 

 tions which exceed the fourth degree can be generally resolved 

 otherwise than by approximation, those of the 6th, 7th, 8th, 

 or still higher degrees, which occur in these solutions, must 

 still require the employment of conjectural trials for obtaining 

 the results to which they lead. ItHs probably for these reasons 

 that practical astronomers have seldom employed any of them, 

 except perhaps that of Laplace, but have adhered to the ancient 

 methods of construction and calculation, which, notwithstand- 

 ing their prolixity, they appear to have found more convenient. 



§ 18. 



In fact, the indirect nature of a calculation is by no means a 

 reason for rejecting its employment ; the only real inconvenience 

 in the present instance arises from the multiplicity of trials 

 required, and from the prolixity of the computations concerned 

 in them ; but the practical astronomer may often have reason to 

 prefer an easier indirect method to a more elegant direct solu- 

 tion of the same problem. Even Laplace has substituted, in 

 effect, for his direct method, a more practicable one, which is 



indirect. 



§ 19. 



The value of any mode of computation must be in the joint 

 proportion of its conciseness, and of the accuracy of the result, 

 on which the facility of the ulterior operations must depend. If 

 the method described in the third section be appreciated upon 

 this principle, I flatter myself that it will be considered as de- 

 serving the preference above all others. But Ave must, in the 

 first place, examine the equations of the first and second de- 

 gree, which have been proposed for the solution of the problem, 

 since, if they were really applicable to the purpose intended, 

 they would certainly afford the simplest and most convenient 

 method of determining the orbit, and would supersede the 

 necessity of inquiring for a new one. 



(To be contwued.) 



