448 Intelligence and Miscellaneous Articles. 



of n quantities cannot exceed 



(n-l)x(n-2).. 3x2x1. 



XIX. Further, in examining these latter values we may con- 

 fine our attention to any one of the functions p, p (r) for instance, 

 aud consider y r as fixed. The determination of the number of 

 values of an n-ary hyposymmetric function is, in fact, reduced 

 to the corresponding determination of an (n— l)-ary function. 



XX. The functions U 4 and u are hyposymmetric, and 



or(w) = 2. ^Vs + 22.^2^8- 

 And, if we make 



yr-y b =Zr, 



we find 



U 5 = Z l Z 3 ?4 + Z<2 ZyZ 3 + £3 ^2^4 + Z 4 Z \ Z <V 



and (see VIII. and XVII.) 



XXI. If all the values of w are known, w' may be made to 

 vanish by means of an equation of the mth degree in y 1 (XVI.). 

 Were we to call the function of z the reducee of that of y, it would 

 be well to inquire whether any relations of value or (hypo- or 

 un-) symmetry exist among successive reducees. The relations 

 between epimetry and hyposymmetry^would be another topic of 

 inquiry. The hyposymmetry of the pure epimetric u would no 

 longer subsist if in a r (y) we replaced y r 2 by y s r , s being greater 

 than 2. Were s equal to 3, cr{u) would consist of a symmetric 

 and a hyposymmetric portion. 



Temple, November 3, 1853. 



LXX. Intelligence and Miscellaneous Articles. 



REMARKS ON LUNAR CIRCLES. BY A. QUETELET. 



DURING the evening of the 19th of January 1853, towards half- 

 past eight, the moon being nearly 60° above the horizon, 

 M. Bouvy, Assistant at the Observatory, saw at the north of the 

 moon two circular arcs of a white colour, the one on this side and 

 the other beyond the zenith. The former was concentric with the 

 moon and about 22° in radius ; the interior part was of a red colour. 

 r Irie second, parallel to the horizon of a very bright silvery white 

 colour, and not iridescent ; it would have been passed by the moon 

 if it had been complete. 



Although this phenomenon belongs to the class of halos, and these 

 circles are described in special treatises, the former under the name 

 of the concentric circle, and the second under that of the paraselenic 

 circle, they nevertheless presented an appearance sufficiently sin- 

 gular to be remarked. 



The moon was at the moment at a considerable elevation above 

 the horizon ; the diameter of the paraselenic semicircle was not 



