202 ON FOBCE8 OF ATTRACTION 



which show the connection that the transverse axis re- 

 maining unchanged has with the permanency of the sys- 

 tem. 



13. He further remarks upon another consequence of the 

 conjugate axis, or the form of the orbit, not affecting the 

 time ; namely, that the conjugate wholly disappearing, and 

 the orbit becoming rectilinear, the theorem applies to the 

 time of falling to the centre, on the centrifugal force or that of 

 projection ceasing to act. (Berlin Mem. 1778.) But Newton's 

 Vlth Lemma, to which he does not refer, in some degree 

 anticipated this also. 



14. The great difficulty of the problem of several centres 

 has been stated. Euler was clearly of this opinion, and he 

 was the first that undertook the solution. After speaking of 

 the general problem (Berlin Mem. 1760, p. 228) as alike 

 important and difficult, he confines himself to the case of two 

 bodies in fixed positions, acting upon a third, which moves in 

 the plane of those distiirbing bodies ; in a word, to the 

 motion of a body drawn towards two fixed centres. He says 

 that, whoever undertakes the solution of this less difficult 

 problem "will find difficulties almost as insurmountable as 

 in the great fundamental problem of astronomy ;" and adds 

 that, after making many fruitless attempts, he had at last 

 been led to a solution by the accident of an error into which 

 he had fallen in his investigation. What he proposes is to 

 find the cases in which the curve is algebraical ; there being, 

 according to the conditions, an infinite variety, most of them 

 transcendental. He considers, however, that if this case of 

 two bodies in fixed centres, and in the same plane with the 

 body attracted, should be incapable of solution, the general 

 problem must prove still more so. Nothing can exceed the 

 clearness of his investigation ; and the ingenious subtlety of 

 the contrivances by which he facilitates the reduction of his 

 differential equations to those of a lower degree. Of this 

 Lagrange expresses great admiration, who, in giving a solu- 

 tion of the case in some respects more extended, but in others 

 less, became fully sensible of the difficulties of the process, 



