224 Royal Society, 



function must be connected in some way with the process of copula- 

 tion. 



He also adverts to the necessity of examining the round ligament 

 by the microscope in glycerine in preference to any other fluid ; as 

 this substance renders the cellular tissue mixed with the fibres more 

 transparent without diminishing the distinctness of their character- 

 istic markings. The author next states his views on the changes 

 which take place in the uterus during utero-gestation, and observes, 

 first, that there is no similarity between the fibres of the round liga- 

 ment and those of the unimpregnated uterus, the latter being made 

 up of spindle-shaped nucleated fibres, contained in a matrix of 

 exceedingly coherent granular matter ; that these fibres are best 

 examined in portions which have been broken up by needles, in 

 preference to thin sections ; and that this tissue is well seen in the 

 larger mammals, as in the Cow, &c. In the impregnated uterus 

 the fibres are found much increased in size and distinctness, but 

 devoid of nuclei and comparatively loosely connected ; and the en- 

 largement of these fibres is of itself sufficient to account for the 

 increased volume of the gravid uterus, without supposing that a set 

 of muscular fibres are formed in it de novo. 



Hence he reasons that the unimpregnated uterus consists probably 

 of little more than an assemblage of embryonic nucleated fibres, in- 

 active until the ovum is received into it, after which their develop- 

 ment commences and continues simultaneously and progressively 

 with that of the foetus ; so that when this last has arrived at a state 

 requiring to be expelled, the uterus has acquired its greatest expulsory 

 power. Lastly, the author observes, since the fully-developed fibres 

 cannot return to their former embryonic condition, they necessarily 

 become absorbed, and a new set of embryonic fibres are formed for 

 the next ovum, so that each foetus is furnished with its own set of 

 expulsory fibres; which view is in perfect accordance with the 

 statements of Drs. Sharpey and Weber, with regard to the mem- 

 brana decidua. 



April 18. — "On the Solution of Linear Differential Equations." 

 By the Rev. Brice Bronwin, M.A. Communicated by S. Hunter 

 Christie, Esq., Sec. R.S. 



The methods employed in this paper to effect the solution or re- 

 duction of linear differential equations consist of certain peculiar 

 transformations, and each particular class of equations is transformed 

 by a distinct process peculiarly its own. The reduction is effected 

 by means of certain general theorems in the calculus of operations. 



The terms which form the first member of the first class of equa- 

 tions are functions of the symbols w and r, the latter being a function 

 of x, and the former a function of x and D, x being the independent 

 variable. This member of the equations contains two arbitrary 

 functions of sr, and may therefore be of any order whatever. It 

 likewise contains two simple factors, such for example as zet and 

 vr-i-nk, which factors are taken away by the transformation em- 

 ployed, and consequently the equation is reduced an order lower; 

 it is therefore integrated when of the second order. There is a se- 



