322 PROCEEDINGS OF THE AMERICAN ACADEMY 



renders it less accordant than Zollner's own formula with the results 

 of his observations. Upon examining the residuals, I find the results 

 to be as follows : — 



Sum of Sum of a veriee 



Positive Ne-'iitive ,f^^^'^^^ 

 Residuals. Residuals. I'^vation. 



By Zollner's formula, 0.326 0.314 0.029 



By equations (2) and (6), when v = 60°, 0.166 0.648 0.037 



By assuming L to increase 0.137 for an I a -io 4"1 04^ 

 increase of 10° in v, J 



It appears, then, that Zijllner's formula probably represents the 

 actual phases of the Moon, near opposition, better than it represents 

 the phases of a furrowed cylinder, computed upon Lambert's hypoth- 

 esis. The similarity between the two objects is certainly not great 

 enough to make it at all surprising that their phases should diifer. 



It may now be asked whether any useful inference can be derived 

 from the theoretical investigation undertaken by ZiiUner, and re- 

 examined in the preceding pages. Before deciding this question in 

 the negative, it may be well to review the facts which have hitherto 

 come separately to our notice. 



In the first place, while Lambert's hypothesis of the emanation of 

 light stands greatly in need of experimental verification, it is still too 

 accordant with some familiar facts to be regarded as a mere conjecture. 

 Smooth rounded surfaces under a diffused illumination (as, for ex- 

 ample, drifts of snow under a cloudy sky) have a general appearance 

 of uniform brightness, which makes it difficult to distinguish their out- 

 lines against a background of the same color. It may well be that 

 the true law of the emanation of light is not so simple as Lambert 

 regarded it, while yet his hypothesis may furnish a tolei-able approxi- 

 mation to the truth. The corresponding hypothesis with regard to 

 incident light is so universally accepted, that it needs no verbal sup- 

 port, although additional experimental evidence with regard to it, and 

 especially with regard to the modifications which it may need in 

 practice, is highly desirable. 



Accepting these two hypotheses, Lambert's formula for the phases 

 of a smooth distant sphere follows as a matter of course ; and the 

 attempt has been made above to show that any smooth surface of revo- 

 lution, with its axis perpendicular to the plane determined by the 

 positions of the source of light, tlie illuminated body, and the observer, 

 will resemble the sphere in its phases. It is therefore somewhat 

 surprising to find that the actual phases of the Moon, at least near 

 opposition, disagree so decidedly with the results of Lambert's formula. 



