08 Mr. J. C. Adams on Professor Challis's new Theoi'ems 



unfavourable judgement led to this withdrawal, had of his own 

 accord communicated to him some of the reasons on which this 

 judgement was based. Professor Challis, however, thinks these 

 reasons to be very unsatisfactory, and consequently invites the 

 reporter to discuss with him the questions on which , they are at 

 issue, in the pages of the Philosophical Magazine. ,..,., .,^ 



As I am the reporter thus referred to, I beg that you will 

 allow me to state some reasons which appear to me sufficient to 

 prove, beyond a doubt, that the principal conclusions of Professor 

 Challis's paper are erroneous, in order that he may have the 

 opportunity, which he desires, of replying publicly to my object 

 tions*. At the same time, 1 must decline to enter into any pro- 

 longed 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 pro- 

 cess, on the ground that the introduction of the quantities usually 

 denoted by c and g into the first approximation 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 num^lpi,^^ 

 of arbitrary constants to fulfill any given initial conditions, cj-vfj^ 



Any process which does this, no matter how it may be 5if^- 

 gested 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 ori- 

 ginal differential equations, is not really the law of nature. 



If we begin with the supposition that the moon's orbit is an 

 immoveable ellipse, the differential equations cannot be satisfied, 

 without adding, to the fii-st 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. 



* It may be proper to mention that the opinion of the other reporter on 

 the paper perfectly agreed \^ith my own. 



