FUNDAMENTAL EQUATIONS. 



cclxi 



The Cambridge additional equations for semidiameter gave the values 



t = 8".583, u = + 5".823, 

 or, after putting u = 0, 



t= + 0".V245, 



which latter value may be substituted in the Cambridge equations. 



Finally, in order to reduce the number of unknown quantities still farther, if possible, and 

 for the sake of a general survey of the character of our materials, all the equations, excepting 

 those derived from the Athens observations, were combined into one set containing seven 

 unknown quantities, by simple addition of the analogous equations in each system. The 

 solution afforded an indication of the admirable precision of Mr. Maclear's measurement of his 

 micrometer-screw, the quantity v 2 (100 times the correction to his adopted value) coming out 

 as zero. The resultant values of x, y, z and w being substituted in the equation containing 

 \_ff] in the Cape Equatorial series, this equation becomes 



1509".463 v 2 = + 0".402 



V 2 = + 0". 000266 



which authorizes us to dispense with any farther consideration of this term. 



For the Santiago Equatorial this solution gave Vi = + 0".5TO 



for the Greenwich Circle, t = + 0".166 



and for the Athens -Circle, t = + 1".927. 



Considering these last quantities as still undetermined, we hare but seven unknown 

 quantities remaining, and our equations containing w assume the annexed form, which we 

 may consider fundamental. 



FUNDAMENTAL EQUATIONS FOR MAES I. 



Santiago Kquatori.il . . . 

 V\'a.sliin^Iiin Kqiiiitoiial . . 

 Cape of Good Hope Equatorial 

 Cape of Good Hope Meridian 

 (!n rmvieh Equatorial . . . 

 Grernwicli Meridian . . . 

 Cambridge Equatorial 

 Athens .Meridian 



245.9801+ 50.301 y + 66.0121 



93.950 x 92.428 y + 100.243 z 



15<i.410 x 114.932 y -f- 75.039 z 



46. 000 x 29.619y + 21.6941 



20.200 x 5.2U1 y + 4.297 z 



24. 000 x -f 4.677y + 26.950 z + 17.000 t, 



60. 080 x 59.5U3y + 47.431 z 



38.000X 1.865y + 21.565Z+ 0.000t 2 



65.862V +322. 302 w +195.44 =0 



36. 292 w +329.73 =0 

 + 221.511 w +510.68 =0 

 + 64. 792 w +130.97 =0 



16.548 w 1.00 =0 



14. 584 w 11. =0 



37.988 w + 176.26 =0 



11. 854 w +257.30 =0 



Santiago Equatorial .... 



\\ ;t^hiiigton Equatorial . . . 

 Cape of Good Hope Equatorial 

 Cape of Good llcpi- Meridian 

 Greenwich Eqint'inal . . . 

 Greenwieli Meridian . . . 

 Cambridge Equaluniil . . . 

 Athens Meridian 



50.301 X + 165.077 y + 3U.811 z -J 



92.428 X + 250.553 y 211.748 z 

 114.932 X + 187.576 y 103.827 z 

 29. 619 X + 54. 230 y 30.617 z 

 5.291X+ 10. 749 y 5.903 z 

 4.677X+ 67.3S9y+ 8.152Z+ 2.944t, 

 . 59.593 x + 118.579 y 96.401 z 



1.SJG5X+ 53. 921 y 0.005Z+ 0.097f 3 



10.342 V + 39. 760 w 470.31 =0 



+ 39.342 IV 616.86 =0 



169.220 w 561.35 =0 



43.931 w 170.00 =0 

 + 5 911 w 16.64 =0 

 + 0.894 w 139.70 =0 

 + 41.064 w 277.99 = U 

 + 2.379 w 170.98 =0 



) Equatorial . . .' . 

 W'l.-iiiu^tuii Equatorial . 

 Cape of G"(id Mope Equatorial 

 C ipe of Good Hope Meridian . 

 Greenwich Kquatnrial 

 Greenwieli Meridian . . 



Athens Meridian 



C6.012X + 9.811 y + 32.841 z 



100.243 x 211.748 y + 222.347 z 



75. MSI x 103.827 y + 69.424 z 



21.GU4X 30.617y+ 21. 778 z 



4.297x 5.903y + 3.290 z 



26.950X+ 8.15'iy+ 58.878Z+ 9.015t, 



47.431 x 96.401 y + 80.976 z 



21.565X O.OOSy + 22.935Z 2.405t 2 



!v+ 80. 599 w 48.93 =0 



40. 298 w +498.27 =0 

 + 106. 929 w +316.13 =0 

 + 3J.783W+ 93.80 =0 



4.382 w + 5.02 =0 



14.748 w 64.67 =0 



31. 783 w +220.35 =0 



6.502W +131.85 =0 



