264 



CATALOGUE OF POLAR STAES. 



Initial Function. 



d a a 

 dt 3 



for w 



8 



10 



16 



d* a 

 lit* 



for io 



8 



w 



16 



d*cc 

 dt* 



for to 



8 



w 



16 



jtxi 



Logarithms. 



«. 



+ 0.000000000000000000000272 



8.43457 



+ 



63 



7.79934 



+ 



125 



8.09691 



+ 



119 



8.07555 



U 



xn 



0.00000000000000000000000112 6.04922» 



80 5.90309/1 



It will be advantageous to divide 



the computation of the 



U 



into two parts. 



U n t 



Eepresenting the sum of the first five terms of the series by JT 4 , we have 



a = a + XPt+ U n # + U™? + U tv f= T if 



in which the terms U 1 , U u , U m , U 1V , are those marked with an asterisk on 

 pp. 260, 261. F 4 will now remain unchanged, whatever the results for the higher 

 powers of the time. 



We shall have, therefore, 



a= r 4 + U v t? + TT^fi + trnfy&Q. 



The computation of the terms U l t, U ll t\ U m t\ U 1Y t\ may be advantageously 

 performed by the summation of the series of differences derived from the differ- 

 ential coefficients. 



For the 



U we have 







A 



d 



a 



dt ' 



A, 



d*a 

 dt 1 ' 



A 



3 



2.3 



d*a 

 dt 9 ' 



*4 



2.3.4 



d*a 

 dt* ' 



&c. 



