TRANSACTIONS OF SECTION A. 4.61 
models illustrating the rotation of a four-dimensional body about a plane by the 
sections of it, with a space containing that plane. Professor Schoute showed 
some lantern-slides in connection with the subject. 
4. Models of Three Developable Surfaces. By Professor ScHourts. 
The author showed three models of developable surfaces in connection with 
the equations 
uw + dura + duy+2=0, 
us + 6urx + 4uy+2=0, 
uo —1but + lbu?x + Guy +2=0. 
He moreover indicated that, if only the equation 
u"+ Ayu 1+ Ajut34+ ... +Anw+A,=0 
has all its roots positive, the equation 
un + Ayu 24 Aw —4e 2. + An qut+au?+yu+z=0 
may represent all possible cases of 2n, 2n—2,... 2,0 real roots, and that by 
means of the double curve the corresponding developable surface really must, and 
will, divide space into z+ 1 regions. 
5. On an Unrecorded and Remarkable Feature in the Splash of a Drop. 
By Professor A. M. Wortuineton, F.2.S. 
The object of this paper was to call attention to the fact that the impact of a 
drop excavates a perfectly spherical hollow, which reaches its greatest depth at 
apparently the same time that the water thrown up attains its maximum height. 
The volume of this spherical pit is enormously greater than the volume of the 
drop, being 360 times as great with a height of fall of 177 cm., forty-four times es 
great with a height of fall of 40 cm. The spherical hollow is lined by the 
original liquid of the drop in the form of a thin layer. The centre of the sphere 
descends as the radius increases till a maximum depth is attained. 
The phenomenon appears to be one in which surface tension plays but a small 
part, and should be capable of hydrodynamical treatment. 
6. A Property of Abelian Groups. By Haroip Hitow. 
Let 24) ty.» +) Um tyy toy. . ., fm be two sets of positive integers or zeros 
such that m+ Umit. +. +Usdtmttnot... +t, for all values of s between 
1 and m inclusive. 
Denote tpt mat... +Us—ta—tma— > - —ter1 by halkm=Um), tm + tn-1 
+...+t, byt kytkhpit... +h, by &. 
We shall write f(z) for (p"—1)(p"1-1).. .(p?-1)(p—1) where p is a 
prime, and suppose /(0) =1. 
Take the coefficient of 
yo in (l+py)(1+p’y)... (L+p%y), of 
y2 in (1+ ph*ty)(1+ phy)... (L+ph*y), of 
yein (L+phthtly) ... (Ltphttthy), .. ., of 
y'n in (La pits may) het (La pits ss thm—rHing, 
and form their product P. We thus obtain part of the coefficient of 
y' in (1 +py)(1+p’y) ... (L+ph* °° *'"y); 
