46 ROYAL SOCIETY OF CANADA 



(2) With regard to the effect of the solar acceleration on the 

 synodic month, it is to be noted that much the greater part of the 

 decrease in the tropical year is due to an increasing rate in the preces- 

 sion of the equinoxes. This does not affect the length of the synodic 

 month, which depends upon the relative motion of the sun and moon. 

 The small remaining part of the solar acceleration would only affect 

 the sj'Dodic month to the extent of ^ of one per cent of its whole rate 

 of decrease ; which would be compensated by a very few y^ths of a 

 second in angular value in the secular acceleration of the moon. This 

 amount has therefore been neglected, in view of the uncertainty of 

 several whole seconds still outstanding between the best determinations. 



(3) The remaining assumption is the one made throughout; that 

 the length of the mean solar day has been constant during the past 

 twenty-five centuries. On this we may remark, as we are now dealing 

 with very minute variations, that astronomers consider that any change 

 is likely to be a retardation, owihg to the friction of the tides; while 

 geologists take the view that the continuous cooling of the earth may 

 still occasion shrinkage, which would tend towards acceleration in the 

 earth's* rotation. It is also to be noted that two of the three cycles 

 we are now considering, furnish ratios between the lunar and solar years ; 

 and being ratios, they would not be affected by change in the unit length 

 of time by which both of these years are measured. 



Data and authorities. — The solar data are from the Annuaire, 

 Bureau des Longitudes, which gives the concordant determinations made 

 by Hansen and Leverrier; and we give also ISTewcomb's more recent 

 determination. The lunar data are from the " Encyclopaedia Britan- 

 nica,'' and from " Elements of Astronomy," by Sir Eobert S. Ball, in 

 the Text-books of Science series. In these, all the leading determinations 

 of the secular acceleration of the moon are summarized. We have also 

 our former acknowledgments to repeat, with regard to direct inform- 

 ation. 



Tropical Year. Variation in length, per century : — 



Annu.aire for 1904 0.539 second = 0.000 006 24 of a day. 



Newcomb's value 0.530 " =0.000 008 14 



Synodic Month. The following summary gives the various deter- 

 minations of the amount of the moon's secular acceleration, with tlieir 

 authorities. Hansen's large value is stated by Newcomb to be theo- 

 retically erroneous; because in computing in what manner the eccen- 

 tricity of the earth's orbit enters into the moon's motion, he took acrount 

 only of the first approximation, as Laplace also had done. We have 

 adopted limiting values as indicated, to correspond with the range in 

 the determinations which mav be considered the most trnstworthv. If 



