758 Mr. Herbert Hall Turner [Feb. 18, 



the attraction of the earth which we formerly asrsumed, and hence 

 our first calculations must be sensibly altered. Without following 

 the difficulties into detail, we can see how they arise, and how they 

 will continue to arise if we look at the problem in this way. We 

 might call the first orbit 0^, calculated with the earth alone ; in the 

 second, Og, we should consider the sun' s attraction ; but now we 

 must go back and correct the earth's attraction, which is seriously 

 modified even by the slight change from 0^ to O2, and we get a new 

 orbit O3 ; then we must correct the sun's attraction, and so on. This 

 is not the actual way in which the calculations are made, but it 

 suffices to illustrate some prominent features of the work, and especi- 

 ally that it is conducted by continued approximations. It will 

 suggest also, that if, instead of being always chiefly under the attrac- 

 tion of one of the iDodies (the earth) and only slightly disturbed by 

 the other — if instead of this the control were more equably divided, 

 the difficulties would be increased. Eecently a satellite of Jupiter 

 (the eighth) was discovered, which is so far from Jupiter, that his 

 control is not much more decisive than that of the sun, and the 

 difficulties were so enormously increased that special methods of 

 calculation had to be devised. The notion of determining a complete 

 orbit round Jupiter first of all had to be given up entirely ; it was 

 necessary to follow the satelhte step by step, assessing at each point 

 the exact attraction of the sun and Jupiter for a little distance ahead, 

 verifying that the satellite would then move in the estimated way, 

 correcting the assumption for any discrepancy found until forecast 

 agreed with result, and so making each point in the path secure 

 before venturing to the next. The process required great labour, but 

 the labour was not grudged in the hope of success, and the hopes 

 were justified. After an interval of some months, during which it 

 was lost in the glare of the sun's rays, the tiny satellite was found 

 a,2:ain close to the place which these laborious calculations had found 

 for it ; and it is, perhaps, not too much to say, that in no other way 

 at present known could this satisfactory result have been attained. 



The problem of H alley's comet is somewhat different. Its 

 movements are chiefly determined by the sun's attraction alone, but 

 there are occasions when it passes near Jupiter or one of the other 

 large planets, when for a brief period the disturbance of the planet is 

 of enhanced importance, and when the methods just described are 

 suitable. Indeed they are suitable generally except for the draw- 

 back of the labour involved, and we have learnt by long experience 

 in astronomy, that when there are two alternative procedures, one 

 of which involves more labour but seems to promise rather greater 

 accuracy, it is often a real saving of time in the end to face the 

 labour at once. This at any rate was the decision at which Messrs. 

 Cowell and Crommelin arrived ; they determined to apply to Halley's 

 comet the procedure found successful in the case of Jupiter's eighth 

 satellite, and after an herculean piece of computation, obtained such 



