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ethyl caprylate and capronate in small quantities and to judge from 
the b.p. also a little of the methyl esters of the same acids. 
The crystals separated from the oil were recrystallised from boiling 
alcohol in which they are but sparingly soluble. The m.p. is 60°. 
The elementary analysis gave a result showing that this substance 
is a saturated hydrocarbon (found C 85.2 H 15.2). 
Sulphuric acid has no action, even on being heated; its solution in 
chloroform does not decolorise a bromine solution, whilst potassium 
permanganate in acetone does not react. 
Parraflins, m.p. 55°—65°, not unfrequently occur in essential oils, 
on the other hand n. capronic- and n. caprylic acid but rarely occur 
in the free state in such oils and then only in minute quantities. 
The essential oil from the fruits of Morinda dtrifolia which consists 
ot more than 90 % of these acids is, therefore, very remarkable in 
that respect. 
Utrecht, Org. chem. Lab. University. 
Mathematics. — “On pentaspheric geometry ” By Dr. S. L. van 
Oss. (Communicated by Prof. P. H. Schoute.) 
1. Referred to a rectangular simplex of coordinates OX l X i X t X A 
and to an imaginary unity point whose cartesian coordinates are! 
x i = x * = = # 4 = i, the equation 
+ * Y +*\ + «*4 + « 1 .==0 .( 1 ) 
represents a real hypersphere H in fourdimensional space S 4 . 
Let X be a point of H, X' the stereographic projection of X on 
~ 0* then, if x, y, z, be the non homogeneous coordinates of X' 
referred to the axes OX, OT, OZ, coinciding respectively with 
OX x , OX 2 , OX t , we have the following relations : 
* a : : ar 4 : = 2.x : 2y : 2z : x* + y* + _ 1 ; **+/+*’+! 
. Putting 
, 2 y=x\ , 2*=*', , *»+**+, *‘+y’+*» + l =t V t , 
we have identically 
*** + ***« + = 0.(1) 
These x'i are known to represent a set of pentaspheric coordinates 
referring A' to the five orthogonal spheres: 
1 = 0 . y = o , , = 0, ** + y* + *’ — 1 = 0 , + + + 1 = 
2 * 
