&$0 Royal Society* 



In relation to physiology, the author submits a new theory of the 

 mechanism of circulation and respiration in the general group of 

 Rotifers, a subject which is but obscurely treated of by the great 

 German observer, who appears to have believed in the existence of 

 tubular vessels or true vascular system. The author thinks, how- 

 ever, that these functions are performed in a manner more resem- 

 bling that of insects, viz. that the blood is contained in the general 

 cavity of the animal and circulates round the lung, which is here 

 represented by a contractile vesicle that receives and expels the 

 water in which the animalcule lives, and so comes to be in interme- 

 diate relation with the air mixed with the water. The difference 

 therefore between the aeration of the blood of insects and that of this 

 rotifer is rather due to the difference of the media they respectively 

 inhabit, than of design. In both, the blood is contained in a general 

 cavity and brought in contact with the air, without the intervention 

 of any true vascular system. 



The beautiful transparency of the animal, and the facility with 

 which the development of the ovum may be traced through all its 

 stages, induces the author to believe it to be well-suited to the in- 

 quiries of the embryologist and of those who devote themselves to 

 the study of the metamorphosis of cell into tissue. 



This animalcule has hitherto been discovered only in a few situa- 

 tions (in Norfolk near Norwich, and in Warwickshire near Coven- 

 try), but it is believed, from the vei'y general dispersion of Infusoria, 

 that it may be more extensively met with, especially in the months 

 of June, July, August and beginning of September. 



The author concludes by expressing his belief that re-examina- 

 tion of the whole order of Rotifera is necessary to determine the 

 disposition of the sexes, and to assign them their proper situation in 

 the scale of animated beings. 



"On the Integration of Linear Differential Equations." By the Rev. 

 Brice Bronwin. Communicated by C J. Hargreave, Esq., F.R.S. 



The method chiefly employed in this paper, is analogous to one 

 which the author had previously applied (Camb. Math. Journal, 

 No. 4) to the integration of such equations in cases where the co- 

 efficients are integer functions of the independent variable. Here 

 they are any functions of that variable, it being however understood 

 that in all integrable cases there must be some relation among these 

 coefficients. The integration is effected by a general theorem of 

 the form 



where D denotes any function of x, and ot a function of symbols 

 both of operation and quantity. By means of this theorem, and the 

 substitution m=ct-j ■ss^. . . zHn v, or some other similar one, the equa- 

 tion is either reduced to an integrable form, or to an equation of a 

 lower order ; or, when neither of these objects can be accomplished, 

 the method may be employed to effect a transformation. 



The method applies most readily to equations of the second order ; 

 but may be applied to those of a higher order, the coefficients be- 



