TIIEORBITOFURANUS. 47 



In forming the next ten lines, it will be noticed that the valne of r„ correspond- 

 ing to any vertical column is found « columns to the right. It is therefore 

 necessary to extend the line v two columns at each end. The extension on the 

 right is, however, omitted for want of space. In performing the subtractions it 

 will be convenient to copy the r's again on the lower edge of a horizontal strip of 

 paper, and, in forming the differences )■„— »■,/ to lay the strip above the line of )''s, 

 and u — u' columns to the riglit. 



On the left of each line of differences is written the f^ictor by which that line is 

 to be multiplied. 



The mode of formation of the A'''s is evident from the formula. 



It will be seen that the same computation wliich gives A', gives also A'_„, only 

 the latter belongs u columns to the riglit. 



Each Ii\ and L\ is multiplied in succession by all the A''s which lie below it in 

 the same column, but the product by IQ is to be written u columns to the right, 

 and that by K_^ u colunnis to the left. The sum of the products in any one 



cos 



column gives the coefficient of ,. (i(j -\-i'l') in the development of ?-j^rp. 



(JVg (^ cos -^ 



This quantity being multiplied by "77" = — .1 — we have cos ij-f^, which only 



needs to be multiplied by sec ^ ^ 1.001103 to give ^p. The units of 7V<^p and 

 ^p correspond to the nintli place of decimals. 



All the periodic terms are to be treated in this manner, all the series of values 

 of ]{,. and Z-„ including the constant term, being subjected to the same process. 

 But, when t' and i are botli zero, i' will be infinite. Here we simply omit the 7', 

 treating it as if it were zero. We thus obtain the complete valne of the terms 

 with constant coefficients in r,-f'^p and }^ which are given in the following table. 

 The terms multiplied by tlie time are still to be computed. They are derived from 

 (20), which may be put in the form 



r:-^ = \M\ n^pj^':? - t,lq,M".' \ nt. 



This expression is computed thus: 



in units of the ninth place of decimals. The value of cos i^i^p is obtained from 

 them by multiplying by r\", exactly as in the case of the constant terms. 



