A REPORT—1862. 
are of different lengths, but of the same shape. The length differs owing to circum- 
stances of growth, such as the left margin being next the stem or next a leaflet, 
forming with it a composite leaf. The curvature of the margin has been ascertained 
im many instances to be that of the reciprocal spiral r=5): In some leaves 
the pole of curvature lies on the axis, in others in the body of the leaf, and in others 
entirely outside the leaf. If the leaflets of a composite leaf have this curvature, 
their extreme points lie on a reciprocal spiral (e. g. the horse-chestnut leaf). It is 
probable that more irregular leaves have margins which are merely modifications 
of the reciprocal spiral or other spirals, such as the Lituus ro ‘ 
The growth of a margin may be represented by increments of an are of the 
spiral cut off by an increasing chord or radius vector. By this means may be 
accurately determined the growth of a leaf under given circumstances of soil, tem- 
perature, and moisture. It is only necessary to register the amount of angular 
rotation of the radius vector of the spiral. 
Quaternion Proof of a Theorem of Reciprocity of Curves in Space. 
By Sir Wit11am Rowan Hamitron, LL.D. sc. 
Let ¢ and y be any two vector functions of a scalar variable, and ¢',y', 6", p" 
their derived functions, of the first and second orders. Then each of the two 
systems of equations, in which c is a scalar constant, 4 
(1).... Sép=e, Sp'p=0, Sp"y=0, 
(2) se ee Syo=c, Sy'o=0, Sy"o=0, 
or each of the two vector expressions, 
BY. rea bistaex ees AY) sa gis Nt eo 
| ORE tae (os SEE 
includes the other. 
If then, from any assumed origin, there be drawn lines to represent the recipro- 
cals of the perpendiculars from that point on the osculating planes to a first curve 
of double curvature, those lines will terminate on a second curve, from which we 
can return to the first by a precisely similar process of construction. 
And instead of thus taking the reciprocal of a curve with respect to a sphere, we 
may take it with respect to any surface of the second order, as is probably well 
lnown to geometers, although the author was lately led to perceive it for himself 
by the very simple analysis given above. 
On a certain Class of Linea\Differential Equations. 
By the Rev. Rosert Harrey, F.R.A.S. 
TuEorEM,—From any algebraic equation of the degree n, whereof the coefficients 
are functions of a variable, there may be derived a linear differential equation of the 
order n—1, which will be satisfied by any one of the roots of the given algebraic equa- 
tion. The differential equation so satisfied is called, with respect to the algebraic 
equation, its “ differential resolvent.”’ The connexion of this theorem, which is due 
to Mr. Cockle, with a certain general process for the solution of algebraic equations, 
led the author to consider its application to the two following trinomial forms, viz. 
Yany+(NH=DeHO. cee ween es eee GC) 
y"—ny"—14+(n—1)2=0, ...... leds OPA ee) 
to either of which any equation of the mth degree, when x is not greater than 5, 
can, by the aid of equations of inferior degrees, be reduced. The several differential 
resolyents for the successive cases n=2, 3, 4, 5 were calculated; and by induction 
the general differential resolvents were formed. © Following Professor Boole’s 
symbolical process and using the ordinary factorial notation, that is to say, repre- 
senting 
(n) (n—1) (2-2)... (w—7r+1) 
