130 ON PROFESSOR CHALLIS'S NEW THEOREMS [20 



into any prolonged controversy on the subject, submitting with confidence 

 what I have now to say to those who are competent to form a judgement 

 respecting it. 



The principal results of Professor Challis's paper are embodied in two 

 theorems, which, as already stated, form the subject of an article in the 

 Philosophical Magazine for April last. As my main objections to the paper 

 relate to these theorems, I shall confine my observations almost entirely 

 to the article in question. 



It will be convenient, however, to > make a few preliminary remarks on 

 the nature of the process usually followed in the lunar theory. Professor 

 Challis objects to the logic of this process, on the ground that the intro- 

 duction of the quantities usually denoted by c and g into the first ap- 

 proximation to the Moon's motion is only suggested by observation. He 

 therefore considers the results of the ordinary process to be hypothetical, 

 until they are confirmed by observation. 



But surely the sufficient and the only test of the correctness of any 

 solution is, that it should satisfy the differential equations of motion at the 

 same time that it contains the proper number of arbitrary constants to 

 fulfil any given initial conditions. 



Any process which does this, no matter how it may be suggested to 

 us, must be logical ; and if the results obtained by it should not agree 

 with observation, the conclusion would be that the law of gravitation, which 

 was assumed in forming the original differential equations, is not really the 

 law of nature. 



If we begin with the supposition that the Moon's orbit is an im- 

 moveable ellipse, the differential equations cannot be satisfied, without adding, 

 to the first approximate expressions for the Moon's coordinates, quantities 

 which are capable of indefinite increase ; and this proves, as is stated by 

 Professor Challis, that an immoveable ellipse is not, or rather does not 

 continue to be, an approximation to the real orbit. 



But if we introduce the quantities usually denoted by c and g, having 

 assigned values slightly differing from unity, which amounts to supposing 

 the apse and node to have certain mean motions, we find that the differ- 

 ential equations are satisfied by adding to the first approximate expressions 

 for the Moon's coordinates, terms, which always remain small; and we thus 



