FUNDAMENTAL EQUATIONS. 



6. Greentrich Mn-iilhin. 



S4.000X + 4.677y +9B.850X+ 17.000 t- 14. 584 w- 11.50 =0 



4.677 x + 67.3W)y + 8.158 x + 8.944 t + 0.894 w - 139.70 =0 



96.MOX + 8ll52y + 58.878X + 9.015 t 14.748 w 64.S7 = 



17.0001+ 9.944y + 9.015*+ 19. 000 t 10. 854 w + 8.80 =0 



-14.984x4- 0.884y 14.748. * 10.854 t + B.lSOvr 9.41 



[ii] -=- 380.83 



7. Cambridge Eqvafarici. 



60.0801 58. 593 > + 47. 431 X 0.900t 1.006 U 37.968 W + 176.98 =0 



58.5931 + 11H.S7S y 06.401 m+ 8.803 t + 13.557 u + 41.084 w 877.99 =0 

 47. 431 x 96.401 y + 80.9761 8.774 t 13 796 n 31. 783 w + 990.35 =0 



0.900 x + 8.803 y 8.774 x + 31.020 t + 50.396 a - O.OOSw 19.58 =0 



1.008X+ 13.5S7y 13.796 x + SO. 198 t + 81 .958 n 0.307 w 17.86 = 



37.8881+ 41 -064 y 31.7831 0.009 1 0.307 a + 8T..011 w 190.94 =0 



[n.] = 796.07 



81 .780 t + 35. 167 a 15.78 = 

 35.167 t + 56.844 n 95.84 = 



[IMI] = 90.09 



8. Athens Meridian, 



38.000 x 1. 865 y + 91.565 x + 0.000 t f 0.330 n 11.854 w + 957.30 



1. 865 X+ 53.921 y 0.005 *. + 0.097t+ 0.996 a + 9.379 w 170.98 =0 



91.S65X 0.005 y + 22.9351 9.405t- 3.205 u 8.509 w + 131.85 =0 



0.0001+ 0.097y 9.405 x + 38. 000 t + 58.069V 0.032 w 70.70 =0 



0.330X+ 0.996y 3.205 x + 58.062 t + 90.252 u 0.193 w 107.04 =0 



11.8541+ 9.379y 8.509x 0.032t 0.123+ 3.758W 85.77 =0 



[mi] =9543.616 



We have here to combine eight systems of equations containing in strictness twelve unknown 

 quantities, viz : the x, y, z, w, common to all, the two v's for the micrometers of the Santiago 

 and Cape equatorials, and the different values for both t and u in three several systems. The 

 observations with the Greenwich equatorial were made by pointing to the estimated center ; so 

 that the unknown corrections to the measured semidiameter find no expression in the final 

 corresponding equations. With the Santiago, Washington, and Cape Equatorials, and the 

 Cape Mural, both limbs were regularly observed ; so that the quantities t and u are here fully 

 eliminated. There remain the measures with the Meridian-circles at Greenwich and Athens, 

 and with the Cambridge Equatorial. But the impossibility of eliminating the influence of 

 both t and u from the Greenwich meridian observations was, from the beginning, so palpable 

 that only one has been introduced into the equations of condition. Our unknown quantities 

 are thus reduced to eleven, and, of course, the first object is to diminish this number as far as 

 possible. 



The details and unsatisfactory issue of the attempt to discriminate between the influences 

 which affect the measured diameters will be set forth in another place. For the present pur- 

 pose it will suffice to say that the effort was utterly futile for every one of our planet-series, 

 and that each new endeavor furnished only a new illustration of the impossibility of any such 

 discrimination with the materials now in our hands. Most of the solutions furnished by the 

 additional equations whether these were independently discussed, or taken in connection with 

 the fundamental equations containing terms dependent upon the same values were illusory 

 plays with figures ; and those concerning which such an expression would be too strong were 

 clearly entitled to no confidence. After much labor devoted to this end the quantity u was 

 finally ignored in all the equations, thus diminishing the number of unknown quantities by two. 



