SECT. 3.] ANY NUMBER OF OBSERVATIONS. 253 



not insure the greatest accuracy, if we were to select three or four positions only 

 for the determination of the orbit, and neglect all the rest. But in such a case, 

 if it is proposed to aim at the greatest precision, we shall take care to collect 

 and employ the greatest possible number of accurate places. Then, of course, 

 more data will exist than are required for the determination of the unknown 

 quantities : but all these data will be liable to errors, however small, so that it 

 will generally be impossible to satisfy all perfectly. Now as no reason exists, 

 why, from among those data, we should consider any six as absolutely exact, but 

 since we must assume, rather, upon the principles of probability, that greater or 

 less errors are equally possible in all, promiscuously ; since, moreover, generally 

 speaking, small errors oftener occur than large ones ; it is evident, that an orbit 

 which, while it satisfies precisely the six data, deviates more or less from the 

 others, must be regarded as less consistent with the principles of the calculus of 

 probabilities, than one which, at the same time that it differs a little from those 

 six data, presents so much the better an agreement with the rest. The investiga 

 tion of an orbit having, strictly speaking, the maximum probability, will depend 

 upon a knowledge of the law according to which the probability of errors de 

 creases as the errors increase in magnitude : but that depends upon so many 

 vague and doubtful considerations physiological included which cannot be 

 subjected to calculation, that it is scarcely, and indeed less than scarcely, possible 

 to assign properly a law of this kind in any case of practical astronomy. Never 

 theless, an investigation of the connection between this law and the most prob 

 able orbit, which we will undertake in its utmost generality, is not to be regarded 

 as by any means a barren speculation. 



175. 



To this end let us leave our special problem, and enter upon a very general 

 discussion and one of the most fruitful in every application of the calculus to 

 natural philosophy. Let V, V, V&quot;, etc. be functions of the unknown quantities 

 p, q, r. s, etc., u, the number of those functions, v the number of the unknown 

 quantities ; and let us svippose that the values of the functions found by direct 

 observation are V = M, V = M , V&quot; = M&quot;, etc. Generally speaking, the 



