1891.] clans of inequalities in the moon's motion. 221 



coefficient of the parallactic inequality. Delaunay's series for this 

 inequality is 1 



- % [-^m + $£m 2 + ^-m 3 + 1 % 9 - 7 -m 4 + ^Hf-^m 5 + 63 |g|| 13 m 6 



74"'02 34f"-33 11"'89 4"-43 l"-86 0"-7l 



_L 10835 53 7 1 59 ,»/] 

 ~ 1116608 "* J' 



0"*38 



where the coefficients expressed in seconds of arc are written 

 below. In this expression put m = m'/(l + in) and expand in 

 powers of m, subtract from the result the expansion of 



(igsm! + |m /2 )/(l - 4m<) 



(which are the first two terms found by rectangular co-ordinates) 

 in powers of in and we obtain instead of Delaunay's series the 

 expression 



- % [(¥•»' + f O/a - W) - ftt^ - W#™' 4 + HtfF™' 5 



-118"25 -ll"-47 +l"-83 + 0"-37 -0"-16 



_ 3 1406 5 7 ^/6 , 3320 080 247^/71 

 245 7 6 " v ^ T9 66W '"' J" 



+ 0"-02 - 0"-20 



Newcomb' 2 suggests that the last term of Delaunay's expression 

 is wrong and this expression which is deduced from Delaunay's 

 would seem to support that view. I hope however to verify the 

 term. Leaving this term out of consideration the increased con- 

 vergency of the series is manifest. The first three terms give 

 nearly the whole value of the coefficient. In the paper the other 

 coefficients of the periodic inequalities of this class will be dealt 

 with in a similar manner, and expansions will be given for the 

 whole class of inequalities, the factor 1/(1 —4m' — ...) being intro- 

 duced also into the higher powers of in. It will then not be 

 necessary to go further than m' 5 a/a' to get the values of the 

 coefficients correct to one hundredth of a second of arc; and, for 

 this degree of accuracy, by the methods given the approximations, 

 either for algebraical or numerical results, are not long. 



(2) On Pascal's Hexagram. By H. W. Richmond. 



The author applies Cremona's method of deriving the hexa- 

 gram by projection of the lines on a nodal cubic surface from the 

 node. By use of a new form of the equation to this surface the 

 equations of the lines are obtained in a perfectly symmetrical 

 form, and their properties thence developed. 



1 Memoires de V Academic des Sciences, Tome xxix. p. 847. 



2 Astronomical Papers for use of American Ephemiris, Vol. i. pt. n. p. 71. 



