TRANSACTIONS OF SECTION A. 407 
4. On a Geometrical Illustration of a Dynamical Problem. 
By Professor Rosert 8. Batt, LL.D., F.R.S. 
A rigid system, with freedom of the second order, is able to make small twists 
about a singly infinite number of screws. If each screw be represented by a point, 
then— 
(1) All the points lie on a circle. 
(2) The angle between two screws equals the angle subtended at the circum- 
ference by their corresponding points. 
(3) The pitch of each screw equals the distance of its point from a ray. 
(4) Two reciprocal screws correspond to the extremities of a chord through the 
pole of the ray. 
(5) The impulsive screws and instantaneous screws correspond to the homo- 
graphic systems of points. 
(6) The directive axis of the homography passes through the pole of the 
ray. 
5. On an Approximate Expression for x! By Professor 
A. R. Forsyra. 
By a well-known formula we have 
— B B 
log a! =log V2n + (a+4) log e—«+ 797-374 33 
B,, By, . ... being Bernouilli's numbers, and their values being given by 
By=h Bs = 30) B,=7 see 
so that for large values of 2 we have approximately 
ie sits 
al=V2ne Ql, 
the error being of the order 3, of the product. 
xv 
But an expression somewhat more accurate, and by no means complicated in 
form, can be obtained from the above. It is easy to verify that 
ae ele 
log a=log @ty)—" +3 Fo aE oat nee 
being any quantity less than x: so that we easily obtain 
log 2!=log /2m + (a +4) log (c+ ")—(x +p) 
‘42 B 1 ae 
Ter Car wel eae ae 
LY (ie we B 
es Caer eer ee 
Choose p, which is as yet arbitrary, so as to satisfy 
pe —p+B,=0, 
so that if p,, #. be the two values 
+ 
By +pg=1, Hyp = B, = 3. 
Substitute », and p, in succession in the above series; add the corresponding 
sides together and divide by 2. The expressions on the right-hand side are symme- 
tric functions of », and p,, and can therefore be written down free from all surds. 
Now 1 has been so chosen as to remove the term in Zs and it happens that these 
Z 
values of 2, which are sufficient for the purpose, make the coefficient of 5 zero 
