216 



PROFESSOR FORBES ON THE TEMPERATURE OF THE EARTH. 



The following method of determining the absolute maxima and minima of 

 the temperature curves and the epochs seems to be simpler in its application than 

 those hitherto in use. 



Although the temperature-curves cannot be represented, either altogether or 

 in great part, by parabolas, the summits may always be represented sufficiently 

 acciu'ately by osculating parabolas, Avhich may, of course, be determined from three 

 points of the curve, and that with the less error as these three points approach 

 more nearly to the point of maximum or minimum sought. In the preceding 

 cases, the ordinates of the curve are already calculated for abscissae corresponding 

 to every 30^ It is easy to find, by simple inspection of the Tallies, between Avhich 

 two ordinates the summit of the curve lies. It will necessarily be between those 

 having the greatest values ( + or — ) ; or, if there be two ordinates with the same 

 value, it must be precisely half way between (supposing the portion of the curve 

 to be parabolic). 



Let y, il", be the two greatest ordinates (calcu- 



lated by the formula), and let y" be an ordinate half 



way between them (calculated fi-om the Equations, 



Table XVI.) Then the difference of abscissae M N, 



N 0, is in this case 15°. Let it be more generally m, 



a number always positive. Let V P be the axis of 



the parabola whose position is sought; and let its 



distance from the ordinate ?/", or N P be .r ( + if to the right hand, - if to the 



left). Then, supposing the pai-abola found, and the tangent to the vertex drawn, 



by the property of the curve, 



a . Wm= M'V^ 

 a . N' « = W^ 



a . 0' = O' V 



where a is the parameter. Or, 



«(VP-y) =(,« + x)= ... (1) 

 a(\^P-y')==t- .... (2) 



«(vp-y")=('«-^)' • • ■ (3) 



Subtracting (2) from (1), 



a\y"-y!) = m--\-1mx ... (4) 



Subtracting (2) from (3), 



a{ji"-y'") = m^-2mx ... (5) 



Making y"—y'=k. and y"-y'" = '&, and adding together the last two equations. 



a(A + B)=:2OT2 



2mP 

 ''=ATB 



