( 387 ) 



only in large crystals, and excliisiv^ely after the colourless minerals. 



In the aegirinenephelinesyeniteporphyries from Olivenfontein (145), 

 which are very rich in nepheline, the first crystallized mineral is 

 the apatite; it was followed by small crystals of nepheline and 

 sodalite, still later hy larger crystals of i)erforated nepheline, sodalite 

 and felspar, simultaneously with the enclosed small needles of 

 aegirine ; finally the perforated aegirines could still be formed in 

 large crystals. 



On account of the tardy crystallization of the larger crystals the 

 order of succession of the crystallizations can be studied more easily 

 in these rocks than in their normalgrained equivalents. 



The sieve structures described above, can be distinguished from 

 those of the contactrocks and crystalline schists by the occurrence 

 of exclusively idiomor})hic or rounded inclusions, according to their 

 relative age. From the real plienocrysts of the porphyric rocks the 

 larger crystals here described differ in this respect that the inclusions 

 are not ranged after the laws of crystallization of the enclosing 

 crystal. 



x4s the perforated crystals usually show a perfectly idiomorphic form, 

 we see that the rule according to which the relative age of the 

 crystals in igneous rocks is proportional to their idiomorphism, does 

 not hold good here. 



Mathematics. — "A?i ed'teiulon of the intec/ral theorem of FovRmR." 

 By Mr. .1. Droste. (Communicated by Prof. J. C. Kluyver). 



(Communicated in the meeting of' September 30, 1911). 

 As is known, for an extensive class of functions /{.c) the equation 



y(.«)= Ida ixp{.c,y, a)f(;/)d>/ 



becomes an identity in d', if we put b = — a = co and xp{u:,i/,a) = 

 ^=cosa(,c — I/]; in this way we find the integral theorem of Fourier 

 which can be regarded as a limiting case of the series of Fourier. 



In the theory of the integral equations Hilbert and Schmidt have 

 proved developments in series of which those of Fourier are special 

 cases. The following is a theorem which is in such a mannei- an 

 exteusion of the integral theorem of Folkiki{. 



Let K {.i;,y) be a continuous symmetrical kernel, 7 ^ (./,), • ■ • , 7v (-'')> • • • 

 a complete system of normalized orthogonal fuiiclioiis of thai kernel 

 and belonging to the limits of integration (t and A, and P.j, . . . , P.,, . . . 

 the corresponding roots ("Eigejiwerte"). 



