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but here the greatest masters of analysis can scarcely see their way. 

 Mr. Poisson has given in the Comm. des Temps, for 1826 formulae 

 for the distribution of heat through a metallic ring, and endeavoured 

 to accomodate them to the case of an astronomical circle ; but if I 

 may be permitted to say so, with doubtful success. I cannot refer 

 to his original memoirs, but he seems to have overlooked the cooling 

 effect of the air, as his co-efficient h in the expression of the fixed 

 temperature seems to depend on the magnitude of the circle and its 

 radiating power. Now instruments, such as are constructed in 

 England, are made extremely light, so that every part of them can 

 assume the temperature of the air in contact with it in a very few 

 minutes ; those however constructed by Reichenback, and the artists 

 who imitate him, are exceedingly massive. If then we assume that 

 the temperature of the air determines that of the circle with great 

 rapidity, the former is so unsteady where the least agitation or 

 current exists, that any reasoning which supposes regularity in the 

 latter's distribution must be defective. That I do not exaggerate the 

 rapidity of these changes of instruments, will be acknowledged by 

 those who have observed the microscopes of a circle placed near an 

 aperture which admits a current of air. But on this supposition 

 there are two cases worth notice, because they come at once within 

 the scope of Par. (a) : the first where the temperature of the air and 

 therefore of the instrument increases uniformly with the height ; the 

 second where it is any a function of the height alone. In the first Let 

 T be the temperature of the centre, and T + r that at the highest point 

 of the circle, then that at the point whose angular distance from the 

 lowest point = ip is T — r cos <p. If the arms of the circle were 

 equally heated, they would expand equally, and the figure of 

 the instrument suffer no change; but their lengths are unequal, 

 being as the mean temperature of each arm is T — ^ r cos (p 



