384 Royal Society. 



The evidence I adduced in regard to the more coarsely marked 

 Diatomaceae, as Isthmia, &c, being furnished with depressions and 

 not cells, is, I believe, satisfactory and conclusive ; and this view 

 has been admitted in a paper since read before the Royal Society*. 



A different view has been taken of the nature of the finer mark- 

 ings, as those upon some species of Gyrosigma, by the author of the 

 paper last quoted, as by previous authors ; and the object of this 

 note is to direct attention to the support which the extended view 

 argued for by me in the paper above referred to, viz. that the finer 

 markings also correspond to depressions, derives from analogy. 



The structure of the Diatomaceae, and their modes of reproduc- 

 tion, are, as is well known, remarkable; — so much so, that these 

 organisms have been claimed by botanists as members of the vege- 

 table, and by zoologists as belonging to the animal kingdom. The 

 preponderance of evidence is decidedly in favour of their vegetable 

 nature ; but, be this as it may, they must all be classed together, — 

 they form a perfectly natural family. Hence we have a strong argu- 

 ment in favour of the markings upon their valves being identical, 

 and as these are evidently depressions in the genera and species with 

 coarsely marked valves (Isthmia, &c), we should expect from ana- 

 logy that the same would apply to those with finer markings. And 

 this view receives further support, from the fact, that under varied 

 methods of illumination, corresponding appearances are presented 

 by the markings when viewed by the microscope, from those which 

 are very large, as in Isthmia, through those of moderate and small 

 size, as in the species of Coscinodiscus, down to those in which 

 they are extremely minute, as in the species of Gyrosigma, &c. The 

 angular (triangular or quadrangular) appearance assumed by the 

 markings, arises from the light transmitted through the valves being 

 unequally oblique. This may be readily shown in the more coarsely 

 marked valves (Isthmia, Coscinodiscus), which present the true 

 structural appearance when the light is reflected by the mirror in 

 its ordinary position, and the spurious angular appearance when the 

 light is rendered oblique by moving the mirror to one side. 



" Researches on the Theory of Invariants." By William Spot- 

 tiswoode, M.A., F.R.S. 



Invariants may be regarded from two points of view, the permu- 

 tational and the functional. According to the former they are con- 

 sidered as arising from a process of permutation, according to the 

 latter as derivatives from other functions. In this paper the latter 

 course is adopted ; and the following is an outline of the method : — 



Let «=/(*»?»•• V-' a «A-.'--) 



be any homogeneous function of the degree n of the variables 

 x, y, .., in which a aj3 , a aA ... •• multiplied by their respective mul- 

 tinomial coefficients are the coefficients of x a y p . . , x a ' y p ' .., . . Let 

 x, y, . . undergo any linear transformation, and let /, m, ..,V,m' , .., .. 

 be the coefficients of transformation ; also let 



* Proceedings of the Royal Society, June 15, 1854. 



