4.60 TRANSACTIONS OF SECTION A. 
such as the cooling of a hot body—and it is shown by attempting to differentiate 
a* that a* obeys this law. 
The solution in series of this equation is obtained, viz., y= A exp (Kr), and we 
proceed to study the function exp (7). 
As a*.a!=a*t" is the defining property of a* for other than integral indices, 
we discuss the product exp 2. exp y [proving that multiplication of such series is 
legitimate] and find it equals exp (7 +y). 
Hence in the usual way we deduce that exp («) = {exp (1) }*=e*. 
This is the first ‘proof’ for the expansion of a function in an infinite series 
(except the G.P.) that the student will have met. It is taken before the corre- 
sponding proof for the general binomial series as being easier ; but this should 
now be worked out on similar lines, and the deduction of the exponential from the 
binomial series sketched roughly (without rigour in the details being attempted). 
After practice with exponential functions, the series for log (1 + 2’) is reached ; 
first this is worked out by the method of the calculus, and made familiar to the 
student; then the strict proof is attempted. 
Here two proofs are sketched: integration of the series for — , and re- 
= 
arrangement of the series for (1+.)” and comparison with that for evlee@+”, 
Either of these proofs requires further work in the convergence of series to be 
tackled—the first ‘uniform’ convergence, the second the convergence of double 
series, and perhaps it would be well to omit these proofs except for the best boys. 
To give the strict proof of the series for sin 2 and cos 2’, we must suppose a 
course of work on complex numbers to be taken, either pari passu with the course 
that has been sketched or at this stage. 
We then proceed with complex indices defining a°*'4 as exp {(c+7d)loga 
and find the value of exp (7°) as follows :— 
(i) Yiod exp (tx) = / {exp (tx). exp (-iv)} = V {exp (0)} =1 
*, a value of 6 can be found such that exp (tx) =cos 6+ 7sin 6 
(ii. ,*. differentiating exp (¢x’)idx = (—sin 6 +7 cos 6)d = (cos 6 + isin 8)id0 
.. dv=dé 
*, « =6+ constant 
(iii) Putting @=0 in exp (10+) =cos 6+7sin 6 we see that k= 2inm 
“. cos6+7sin 6=exp (76), which gives the series for cos 6 and sin 8. ' 
If we now show that log (complex number) = log (modulus) + 7 (amplitude) and 
deduce the series for tan~!7 from that for log (1+z), we shall have completed 
the usual course on series, with the exception of Taylor's series for f(a +4), which 
should be discussed at the end of the course, as a generalisation of the previous 
work, the remainder after the mth term of the series being carefully investigated. 
3. On Models of Three-dimensional Sections of Regular Hypersolids in 
Space of Four Dimensions. By Mrs. A. Booue Storr. 
After giving an idea of the four different kinds of axes of a regular four- 
dimensional polytope, and having explained in what manner any of these six 
polytopes may be intersected by a range of parallel spaces normal to any of these 
axes, Mrs. Stott exhibited the different kinds of sections that may be obtained 
by models of cardboard differently coloured, so as to show the position of the 
different regions of bounding bodies with respect to the central axis. She also 
exhibited models illustrating the space-filling properties of a three-dimensional 
section of any set of regular polytopes filling-space of four dimensions. AJso 
1 This is a modification of a proof given by Stolz and Gueimer, and is due to my 
late pupil, Mr. McCleland. 
