TRANSACTIONS OF THE SECTIONS. 5 
by [n]", the differential resolvent of (I.) was found to take the form 
a ad n—-1,,_¢,—1)n-1[ _%_, @ _ 2n—1)*-F n-th, 
n Lez | 1 as ) a Ee aI xv y=[1] [m 1] Lees (A) 
In like manner, the differential resolvent of (II.) was found to be 
n—1 d2-1 d d AEA ig n-1 
n [@ Dee | y—(n—1) (na —n—1) na —2 | gy=[n—1] aie,» (B) 
Every differential resolvent may be regarded under two distinct aspects. It may 
be considered either, first, as giving in its complete integration the solution of the 
algebraic equation from which it has been derived; or, secondly, as itself solvable 
by means of that equation. In the first aspect the author has considered the 
differential equation (A) in a paper entitled “ On the Theory of the Transcendental 
Solution of Algebraic Equations,” just published in the ‘Quarterly Journal of Pure 
and.Applied Mathematics,’ No. 20. In the second aspect every differential resol-- 
vent of an order: higher than the second gives us, at least when the dexter of its 
defining equation vanishes, a new primary form, that is to say, a form not recognized 
as primary in Professor Boole’s theory. And in certain cases in which the dexter 
does not vanish, a comparatively easy transformation will rid the equation of the 
dexter term, and the resulting differential equation will be of a new primary form. 
On the Volumes of Pedal Surfaces. By T, A. Hirst, F.2B.S, 
The pedal surface being the locus of the feet of perpendiculars let fall from any 
point in space, the pedal origin, upon all the tangent planes of a given fixed primi- 
tive surface, will, of course, vary in form as well as in magnitude with the position 
of its origin. If, however, the volume of the pedal be considered as identical with 
that of the space swept by the perpendicular, as the tangent plane assumes all pos- 
sible positions,—a definition which will apply to unclosed as well as to closed 
pedals,—the following two general theorems may be enunciated:—1l. Whatever 
may be the nature of the primitive surface, the origins of pedals of the same 
volume are, in general, situated on a surface of the third order. 2. The primitive 
surface being closed, but in other respects perfectly arbitrary, the origins of pedals 
of constant volume lie on a surface of the second order; and the entire series of - 
such surfaces constitutes a system of concentric, similar, and similarly-placed qua- 
drics, the common centre of all being the origin of the pedal of least volume. 
On the Exact Form and Motion of Waves at and near the Surface of Deep Water. 
By Wri11am Joun Macevorn Ranuine, C.L., LL.D., F.RASS. L. & E. §e. 
The following is a summary of the nature and results of a mathematical inyesti- 
gation, the details of which have been communicated to the Royal Society. 
The investigations of the Astronomer Royal and of Mr. Stokes on the question 
of straight-crested parallel waves in a liquid proceed by approximation, and are 
based on the supposition that the displacements of the particles are small compared 
with the length of a wave. Hence it has been legitimately inferred that the results 
of those investigations, when applied to waves in which the displacements are con- 
siderable as compared with the length of wave, are only approximate. 
In the present paper the author proves that one of those results—viz. that in very 
deep water the particles move with a uniform angular velocity in vertical circles 
whose radii diminish in geometrical progression with increased depth, and conse- 
quently that surfaces of equal pressure, including the upper surface, are trochoidal— 
is an exact solution for all possible displacements, how great soever. 
The trochoidal form of waves was first explicitly described by Mr. Scott Russell ; 
but no demonstration of its exactly fulfilling the cinematical and dynamical condi- 
tions of the question has yet been published, so far as the author knows. 
In ‘A Manual of Applied Mechanics’ (first published in 1858), the author 
stated that the theory of rolling waves might be deduced from that of the positions 
assumed by the surface of a mass of water revolving in a vertical plane about a 
