64 Royal Society. 



Thus any triangular number multiplied by 8 with 1 added is equal 

 to the square of an odd number; or, any square of an uneven 

 number minus 1 is divisible by 8, and the quotient is a triangular 

 number. 



§3. 

 Of the higher powers I will confine myself to one example. 



H. 

 Every fourth power » 4 is the sum of an arithmetical progression 

 of » terms, the first term of which is n 2 , and the difference 2» 9 . 



1 =1 + 



4+12 = 2 4 



9 + 27 + 45 =3* 



16+48 + 80+112 =4< 



25 + 75 + 125 + 150 + 225 =5* 



36+10S+180 + 252 + 324 + 396 =6 4 



This triangle consists of the progressions in (D.) multiplied re- 

 spectively by n, or of those in (A.) multiplied by « s . 



" On the Structure and Functions of the Rostellum in Lislera 

 ovata." By J. D. Hooker, M.D., F.R.S. 



The author first gives an account of the form and structure of the 

 rostellum of List era ovata, and its relation and position to the anther 

 and stigma. He finds that the rostellum is divided by parallel septa 

 (at right angles to the plane of that organ) into a series of longitu- 

 dinally elongated loculi, which gradually taper from the base up- 

 wards, and terminate at two opake cellular spots, one on each side 

 of the apex of the rostellum, towards which latter the loculi also 

 converge. When the flower is fully expanded, these loculi are dis- 

 tended with a viscid grumous fluid, full of chlorophyll granules. 

 Their external walls, and the septa dividing them, are formed of a 

 delicate, transparent tissue, which is cellular at the base and apex of 

 the rostellum only. 



Their grumous contents, when examined at the earliest period of 

 development, present the appearance of opake club-shaped com- 

 pressed bodies, with areolated surfaces ; a form and appearance that 

 may be restored at a later period by coagulating with alcohol. 



At the period of impregnation the slightest irritation of the ros- 

 tellum causes the sudden and forcible discharge of the contents of 

 these loculi (through the rupture of the cellular tissue at the apex 

 of the rostellum) and its protrusion in the form of two viscid glands, 

 which coalesce into one, after which the rostellum rapidly collapses 

 and contracts. 



The pollen-masses, when freed from the anther-case, fall naturally 

 upon the rostellum ; they are retained there by their viscid gland-like 

 contents, and, breaking up, the pollen-grains become (by the con- 

 traction of the rostellum) applied to the subjacent stigmatic surface. 



The author adds remarks on the structure of the rostellum in 



