9. The equations to the circles on DD', FF', referred to BC, BA

as axes are

(m2 - n2)(x2 + y2+ 2xycos B) - 2m2ax - 2m2aycos B+ m2a2 = 0,

(m2 - 12)(x2 + y2+ 2xycosB) - 2m3cxcosB - 2m3cy + m2c2 = 0. (v.)

10. The circle (iii.) cuts BC again in d, so that

Bd

c2(m + n)(n + 1) + a2(n+1)(l + m) − b2(l + m) (m +n)

Cd

́a2(n + 1)(l + m) + b2(l + m)(m + n) − c2(m+n)(n+1)

11. If l=m=n, O is the circumcentre, P the centroid, and (iii) the nine-point circle.

13. If P is the circumcentre, then

1:m: n= cosec2A: cosec2B: cosec2C;

and if P is the orthocentre, then

l:m: n= cotA : cotB: cotC,

and (iii.) is, of course, the nine-point circle.

14. If O is the orthocentre, then

and P is

l: m : n = cos A: cosB: cosC,
asin2A = ẞsin2B = ysin2C.

--

15. If l:m:n=s - a: s − b : s − c, then P is the Gergonne point, and (iii.) is the In-circle.

16. If P is the Symmedian-point, then (iii.) is

2(a2+b2)(b2 + c2)(c2 + a2)Z(aßy) = Z(aa). Σ[bca(b1 + c1 — a1 + λ2)],

where 2 a2b2 + b2c2 + c2u3 as before.

=

17. If n =

: 0, m = a, l=b, (iii.) becomes

(a+b)Z(aßy) = Z(aa). [bcosAa+acosBẞ]

which cuts AB in the points where the bisector of C and the perpendicular from C meet it.

18. If the locus of P is the line

pa+qẞ+ry=0,

then the envelope of DE is found from the equations

It is readily seen to be the in-conic

p2a2 + q2ß2 + r2y2 - 2pqaß +2qrẞy + 2rpya = 0.

The chords of contact are

pa-qẞ+ry=0,

− pa + qß+ry == 0,

pa+qB+ry=0.

In like manner, with the same condition, the envelope of D'E'F' is p2a2+q2ß2 + r2y2 - 2grẞy - 2rpya-2pqab=0.

19. If P, P' are inverse points, and O, O' are given by (l, m, n), (l', m', n'), then we have

a2ll' = b2mm' = c2nn'.

The equation to PP', in the general case, is

all'a[m'n - mn']+ ... + ... = 0

(vi.)

hence, if (vi.) is satisfied, and the line passes through the symmedian

point, we must have

[m'n - mn']=0.

(vii.)

20. If O, O', O" are points such that P, P', P" are collinear, then we have

El'l''mn[m'n' - m'n"] = 0.

21. If P, P' are inverse points, the equation (vii.) is satisfied by the cubic

Second Meeting, December 8th, 1893.

JOHN ALISON, Esq., ex-President, in the Chair.

Note on the number of numbers less than a given number and prime to it.

By Professor STEGGALL.

The following proof of the well-known result, n being any number, a, b, c the different prime factors that singly, or multiply, compose it

sider also the number aN where a is a prime factor of N: then the numbers

P, p+N, p+2N, .. p+(a-1)N

are all less than aN.

They are also all prime to N (and to aN) if p is, and not otherwise; for p has no prime factor of N (and of aN) and N, 2N.. etc., have every prime factor of N (and of aN).

Hence if a is a prime factor of N the number of numbers less than aN and prime to it (sometimes called the totient of aN) is a times the totient of N,

or

(aN)= ap(N).

(1)

Again let b be a prime number not a factor of N; then of the numbers

P, p+N, p+2N, p+(b-1)N

one, and one only, is divisible by b.

Hence as before if p is prime to N, b-1 of the above numbers are prime to bN. Thus if b is a prime non-factor of N the totient of bN is (b-1) times the totient of N, or

Now (a) a-1 and we note that the totient of any prime a is (a1) times that of unity, if we call that of unity one.

We see then that in multiplying the prime factors a p times, b q times, etc., in any order, the totient of the numbers resulting is once (viz., at the first introduction of a new factor a) increased in the ratio a 1, and at every other introduction of a in the ratio a; similarly for b, c, etc.

Hence

$(a2bc"..) = a2¬1b1−1c2¬1 ( a − 1)(b − 1)(c − 1)... × (1)

I had intended to bring before you a simple definition of a ridge line on a surface, and to shortly discuss the equation deduced; but since offering my paper I learnt that Dr M'Cowan had developed very fully the consequences of an equivalent definition; and therefore, lest I should accidentally impair the interest of his paper, I shall leave with him the treatment of the whole subject, a treatment that I believe includes all I had to say.

The models I exhibit were originally made with a view to the presentation in a concrete form of the mathematical conceptions of contour lines, lines of slope, saddle points, indicatrices, and ridges in surfaces. As a minor and secondary object, the educational value of the representation of an actual hill seemed to justify the construction of the model of a real mountain rather than that of any surface derived from merely arbitrary design, or from fixed equation. Besides this, there is a probability that such a model as I show may