JUPITER AND SATURN. 



Ill 



In this way 6, i and D &, ^ have been found. The values of pii 

 ;and piz are easily found by extrapolation, if 5i2,i and 613,1 should 

 "be required. The next step is to find D"'6o, 1 and D"'bi^ 1 to such 

 value of m as may be desired ; we chose w = 8. The equations 



are 



log. = 9.62630129. 



D' &o, 1 =/3^^ [2 D + 1] &o, , i¥ = ~~, 



=/32[ 4D2+ 5D + 2]&,,i 

 =/32 [ 6 D3 + 13 D2 + 12 D + 4] h , 

 =/3- [ 8D^ + 25D3 + 38D- + 28D+8]&oi 

 =/3- [10 D ' +41 D^ + 88 D3 + 104 D2 +64 D +16] 6,,, 1 

 =/3- [12 D'^ +61 D"' + 170 D* + 280 D3 + 272 D2 + 144 D + 32] &01 

 =/3-' [14 D' + 85 D" + 292 D'^ + 620 D^ + 832 D=^ +688 D^ +320 D 

 + 64] Oo, 1 



D8 



and 



D3 



, ,+/32 [ 2D +1] &,, , 

 = D /.,,,+/32[ 4D^^ + 5D]&,, 

 = D2 6, ,+/32[ 6D3 + i3D2 + 8D]&, , 

 =.Ds&, ,+/32 [ 8D^ + 25D3 + 32D2 + i6D]&, , 

 = D^6;,+/32 [ioD'' + 4iD^ + 8oD3 + 8oD-' + 32D]&, , 

 = D5 6,' 1 +/32 [12 D« + 61 D-^ + 160 D^ + 240 D3+ 192 D2 



+'64D]&,,, 

 = D«&, , +/32 [14 D' + 85 D« + 280 D' + 560 D^ + 672 D3 

 + 448D-' + I28DJ6,,, 

 On the right hand side of these equations the symbols of operation 

 have, for convenience in writing and printing, been separated 

 from those of quantity ; thus /?- [2 D + 1] 60 1 is to be read 

 /32 (2 D&„,, +&„,,). 



Before proceeding further, it is advisable to check the quantities 

 already found by means of the relations 



D(D-l)"'&,,,=:aD'" + l bi,, 



or D'" &,,,=aD (D + i)'«-i &J.1 



and this has been done for the table on page 113. 



The rest of the D'" bj^ quantities can be computed easily and 

 •expeditiously with a Brunsviga calculating-machine, as the quan- 

 tities are simple additions, subtractions and short whole-figure 

 :multiplications of the quantities already found. 



The formulae to be used are : 



&a3 = 

 6 1,3 = 



&3,3 = 

 ^4,3 = 

 6.3 = 

 6 6,3 = 



&r,3 = 



&8,3 = 



6<i,3 = 

 ^10. 3 = 

 &11,3 = 



6..,3 = 



D2 + D- o)&,, 



D2 + D- 2)&.,', = (D2 + D- o)&, 

 D2 + D- 6)&;, = (D- + D- 2)6; 

 D-' + D- 12) &;, = (D^^ + D- 6)6.; 

 D2 + D- 20) &-, , = (D->D- 12)63 

 D2 + D- 30) 6,', = (D2 + D- 20)6/ 

 D2 + D- 42) 6-, = (D2 + D- 30)65 

 D2 + D- 56) 6;, = (D-+D- 42) 6j 

 D2 + D- 72) 6,., = (D' + D- 56)6, 

 D-^ + D- 90) 6,„, = (D2 + D- 72)63' 

 D2 + D-II0) 6„, = (D2 + D- 90)6., 

 D2 + D-132) 6,.,\ = (D2 + D-iio) 6,,,' 

 D2 + D-156) 6,,,, = (D2 + D-i32) 6„, 

 D-+D-182) 6i4, = (D2 + D-i56) 6,., 



