SECT. 1.] THREE COMPLETE OBSERVATIONS. 201 



Q f = Q. And conversely it is readily perceived, that if in the result P = P, 

 Q = Q, the double calculation of the elements from both combinations would, if 

 completed, furnish numbers entirely equal, by which, therefore, all three observa 

 tions will be exactly represented, and thus the problem wholly satisfied. But 

 when the result is not P = P, Q = Q, let P P, Q Q be taken for X and Y, 

 if, indeed, P and Q were taken for x and y; it will be still more convenient to put 



log P = x, log Q = y, log F log P = X, log $ log Q = Y. 

 Then the calculation must be repeated with other values of x, y. 



147. 



Properly, indeed, here also, as in the ten methods before given, it would be 

 arbitrary what new values we assume for x and y in the second hypothesis, if 

 only they are not inconsistent with the general conditions developed above ; but 

 yet, since it manifestly is to be considered a great advantage to be able to set out 

 from more accurate values, in this method we should act with but little prudence 

 if we were to adopt the second values rashly, as it were, since it may easily be 

 perceived, from the very nature of the subject, that if the first values of P and Q 

 were affected with slight errors, P and Q themselves would represent much more 

 exact values, svipposing the heliocentric motion to be moderate. &quot;Wherefore, we 

 shall always adopt P and Q themselves for the second values of P and Q, or 

 log P , log Q for the second values of x and y, if log P, log Q are supposed to 

 denote the first values. 



Now, in this second hypothesis, where all the preliminary work exhibited 

 in the formulas 1-20 is to be retained without alteration, the calculation will be 

 undertaken anew in precisely the same manner. That is, first, the angle o 

 will be determined; after that e, r , n ~, &quot;--, , r, ? , r&quot;, / , /, /&quot;. From the dif 

 ference, more or less considerable, between the new values of these quantities 

 and the first, a judgment will easily be formed whether or not it is worth while 

 to compute anew the correction of the times on account of aberration ; in the 

 latter case, the intervals of the times, and therefore the quantities & and 6&quot;, will 

 remain the same as before. Finally, 1], if are derived from /, r, r&quot;,f&quot;, r, r and 



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