Prof. J. J. Sylvester on Spherical Harmonics. 29 



bands extinguish the light distinctly, but are penetrated by 

 narrow bandelets which do not extinguish the light, and which 

 make with the base an angle of about 60°, and appear to be 

 parallel to the faces e z = a*. One yellow crystal (fig. 4), very 

 distinctly twinned, as shown by its very decided reentrant 

 angles, consists of five individuals united in the interior along 

 perfectly irregular surfaces. Each of the members of this 

 twin, except the upper small one, contains both bands parallel 

 to p and bandelets parallel to a? of the white crystals. 



It seems to me that we might retain the name Humite for the 

 orthorhombic crystals of type I., that of chonclrodite for the 

 clinorhombic crystals of type II., and seek a name for the 

 crystals of type III. — clinohumite, until a better be found. 

 It ought, however, to be ascertained if all the crystals from 

 Sweden and from America belong to type II., or if the brown 

 crystals from Kafveltorp alone belong to this type, while the 

 grey or brownish ones from Ladugrufvan and Pargas are of 

 type III. (Edward Dana admits the last two types in Ame- 

 rican crystals). It is, however, evident that there is a close 

 crystallographic and chemical relationship between the second 

 and third types, and that they differ most in their optical pro- 

 perties, although Websky tries to show a chemical difference 

 by means of new formulae, which I declare myself incapable of 

 following. This point will be understood in time ; but what 

 was important was to establish first the undoubted facts, and 

 the non-existence of three types of one and the same species, 

 which had always seemed to me an extraordinary thing, 

 difficult to admit, especially in presence of the holohedrism of 

 the one, and the hemihedrism of the two others. 



XXXVII. Note on Spherical Harmonics. By J. J. Sylvester, 

 F.R.S., Professor of Mathematics at the Johns Hopkins 

 University , Baltimore *. 



IF for a moment we confine our attention to so-called 

 " zonal " harmonics, and affect each element of a uniform 

 spherical shell with a density varying as the product of two 

 such harmonics of unequal degrees, we know that the mass 

 of such shell is zero. A very slight consideration will serve 

 to show that this is tantamount to affirming that if a given 

 spherical surface be charged with a density inversely propor- 

 tional to the product of the distances of each element from 

 two fixed internal points lying in the same radius produced, 

 then the mass of such shell will be a complete function of the 



* Communicated by the Author. 

 U2 



